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case p_linear
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
⊢ Polynomial.map (algebraMap (↥F⟮γ⟯) E) p = C (leadingCoeff h) * (X - C β)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
|
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
⊢ ∀ x ∈ roots (Polynomial.map ιEE' h), x = ιEE' β
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
|
intro x hx
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : x ∈ roots (Polynomial.map ιEE' h)
⊢ x = ιEE' β
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
|
rw [mem_roots_map h_ne_zero] at hx
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
⊢ x = ιEE' β
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
|
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
⊢ ιEE' γ - ιEE' (ιFE c) * x ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
|
have f_root := root_left_of_root_gcd hx
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
f_root : eval₂ ιEE' x (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) = 0
⊢ ιEE' γ - ιEE' (ιFE c) * x ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
|
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
f_root : eval₂ (RingHom.comp ιEE' ιFE) (ιEE' γ - ιEE' (ιFE c) * x) f = 0
⊢ ιEE' γ - ιEE' (ιFE c) * x ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
|
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
hc :
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g),
-(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
⊢ x = ιEE' β
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
|
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
hc :
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g),
-(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
⊢ x ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
|
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
hc :
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g),
-(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
⊢ eval₂ ιEE' x (Polynomial.map ιFE (minpoly F β)) = 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
|
exact root_right_of_root_gcd hx
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
hc : -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (x - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
⊢ x = ιEE' β
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
|
by_contra a
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
hc : -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (x - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
a : ¬x = ιEE' β
⊢ False
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
|
apply hc
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
hc : -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (x - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
a : ¬x = ιEE' β
⊢ -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (x - ιEE' β) = (RingHom.comp ιEE' ιFE) c
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
|
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
hc : -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (x - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
a : ¬x = ιEE' β
⊢ -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) = (RingHom.comp ιEE' ιFE) c * (x - ιEE' β)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
|
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : SplittingField (Polynomial.map ιFE g)
hx : eval₂ ιEE' x h = 0
hc : -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (x - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
a : ¬x = ιEE' β
⊢ -((algebraMap E (SplittingField (Polynomial.map (algebraMap F E) (minpoly F β)))) α +
(algebraMap E (SplittingField (Polynomial.map (algebraMap F E) (minpoly F β)))) ((algebraMap F E) c) *
(algebraMap E (SplittingField (Polynomial.map (algebraMap F E) (minpoly F β)))) β -
(algebraMap E (SplittingField (Polynomial.map (algebraMap F E) (minpoly F β)))) ((algebraMap F E) c) * x -
(algebraMap E (SplittingField (Polynomial.map (algebraMap F E) (minpoly F β)))) α) =
(algebraMap E (SplittingField (Polynomial.map (algebraMap F E) (minpoly F β)))) ((algebraMap F E) c) *
(x - (algebraMap E (SplittingField (Polynomial.map (algebraMap F E) (minpoly F β)))) β)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
|
ring
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case p_linear
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
h_roots : ∀ x ∈ roots (Polynomial.map ιEE' h), x = ιEE' β
⊢ Polynomial.map (algebraMap (↥F⟮γ⟯) E) p = C (leadingCoeff h) * (X - C β)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
|
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case p_linear
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
h_roots : ∀ x ∈ roots (Polynomial.map ιEE' h), x = ιEE' β
⊢ Polynomial.map (algebraMap (↥F⟮γ⟯) E) p = h
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
|
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
h_roots : ∀ x ∈ roots (Polynomial.map ιEE' h), x = ιEE' β
⊢ Polynomial.map (algebraMap (↥F⟮γ⟯) E) p = EuclideanDomain.gcd ?p_linear.p_linear_1 ?p_linear.p_linear_2
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
·
|
dsimp only
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
·
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
h_roots : ∀ x ∈ roots (Polynomial.map ιEE' h), x = ιEE' β
⊢ Polynomial.map (algebraMap (↥F⟮α + c • β⟯) E)
(EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮α + c • β⟯) (minpoly F α))
(C (AdjoinSimple.gen F (α + c • β)) -
C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮α + c • β⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮α + c • β⟯) (minpoly F β))) =
EuclideanDomain.gcd ?p_linear.p_linear_1 ?p_linear.p_linear_2
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
|
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
h_roots : ∀ x ∈ roots (Polynomial.map ιEE' h), x = ιEE' β
⊢ EuclideanDomain.gcd
(Polynomial.map (algebraMap (↥F⟮γ⟯) E)
(comp (Polynomial.map (algebraMap F ↥F⟮α + c • β⟯) (minpoly F α))
(C (AdjoinSimple.gen F (α + c • β)) -
C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮α + c • β⟯) } * X)))
(Polynomial.map (algebraMap (↥F⟮γ⟯) E) (Polynomial.map (algebraMap F ↥F⟮α + c • β⟯) (minpoly F β))) =
h
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
·
|
simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
·
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
(C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
h_roots : ∀ x ∈ roots (Polynomial.map ιEE' h), x = ιEE' β
⊢ EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F E) (minpoly F α))
(C α + C (c • β) -
C
((algebraMap (↥F⟮α + c • β⟯) E)
{ val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮α + c • β⟯) }) *
X))
(Polynomial.map (algebraMap F E) (minpoly F β)) =
EuclideanDomain.gcd
(comp (Polynomial.map (algebraMap F E) (minpoly F α)) (C α + C (c • β) - C ((algebraMap F E) c) * X))
(Polynomial.map (algebraMap F E) (minpoly F β))
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
|
congr
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
|
Mathlib.FieldTheory.PrimitiveElement.103_0.R5HND7n71i1v1rZ
|
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
|
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
|
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case intro.intro.intro
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
|
use α + x • β
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
⊢ F⟮α, β⟯ = F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
|
apply le_antisymm
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
⊢ F⟮α, β⟯ ≤ F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
·
|
rw [adjoin_le_iff]
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
·
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
|
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
|
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
|
simp only [← heq] at αyβ_in_K
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
|
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : α + x • β - (α + y • β) ∈ F⟮α + x • β⟯
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
|
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : α + x • β - (α + y • β) ∈ F⟮α + x • β⟯
⊢ α + x • β - (α + y • β) = (x - y) • β
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by
|
rw [sub_smul]
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : α + x • β - (α + y • β) ∈ F⟮α + x • β⟯
⊢ α + x • β - (α + y • β) = x • β - y • β
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul];
|
abel1
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul];
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : (x - y) • β ∈ F⟮α + x • β⟯
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
|
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : (x - y)⁻¹ • (x - y) • β ∈ F⟮α + x • β⟯
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
|
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : β ∈ F⟮α + x • β⟯
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
|
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : β ∈ F⟮α + x • β⟯
⊢ α ∈ F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
|
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.e'_4
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : β ∈ F⟮α + x • β⟯
⊢ α + x • β - ?m.561975 • β = α
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : β ∈ F⟮α + x • β⟯
⊢ F
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
|
apply add_sub_cancel
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯
αyβ_in_K : α + y • β ∈ F⟮α + x • β⟯
β_in_K : β ∈ F⟮α + x • β⟯
α_in_K : α ∈ F⟮α + x • β⟯
⊢ {α, β} ≤ ↑F⟮α + x • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
|
rintro x (rfl | rfl)
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a.inl
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
x✝ y : F
hneq : x✝ ≠ y
x : E
f : F → IntermediateField F E := fun x_1 => F⟮x + x_1 • β⟯
heq : f x✝ = f y
αxβ_in_K : x + x✝ • β ∈ F⟮x + x✝ • β⟯
αyβ_in_K : x + y • β ∈ F⟮x + x✝ • β⟯
β_in_K : β ∈ F⟮x + x✝ • β⟯
α_in_K : x ∈ F⟮x + x✝ • β⟯
⊢ x ∈ ↑F⟮x + x✝ • β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;>
|
assumption
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;>
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a.inr
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
x✝ y : F
hneq : x✝ ≠ y
x : E
f : F → IntermediateField F E := fun x_1 => F⟮α + x_1 • x⟯
heq : f x✝ = f y
αxβ_in_K : α + x✝ • x ∈ F⟮α + x✝ • x⟯
αyβ_in_K : α + y • x ∈ F⟮α + x✝ • x⟯
β_in_K : x ∈ F⟮α + x✝ • x⟯
α_in_K : α ∈ F⟮α + x✝ • x⟯
⊢ x ∈ ↑F⟮α + x✝ • x⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;>
|
assumption
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;>
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
⊢ F⟮α + x • β⟯ ≤ F⟮α, β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
·
|
rw [adjoin_simple_le_iff]
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
·
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
⊢ α + x • β ∈ F⟮α, β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
|
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
α_in_Fαβ : α ∈ F⟮α, β⟯
⊢ α + x • β ∈ F⟮α, β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
|
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
x y : F
hneq : x ≠ y
heq : f x = f y
α_in_Fαβ : α ∈ F⟮α, β⟯
β_in_Fαβ : β ∈ F⟮α, β⟯
⊢ α + x • β ∈ F⟮α, β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
|
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
|
Mathlib.FieldTheory.PrimitiveElement.181_0.R5HND7n71i1v1rZ
|
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
⊢ ∃ α, F⟮α⟯ = ⊤
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
|
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case inl
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
⊢ ∃ α, F⟮α⟯ = ⊤
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
·
|
let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
·
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case inl
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
⊢ ∃ α, F⟮α⟯ = ⊤
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
|
have base : P ⊥ := ⟨0, adjoin_zero⟩
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case inl
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
base : P ⊥
⊢ ∃ α, F⟮α⟯ = ⊤
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
|
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
base : P ⊥
⊢ ∀ (K : IntermediateField F E) (x : E), P K → P (restrictScalars F (↥K)⟮x⟯)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
|
intro K β hK
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
base : P ⊥
K : IntermediateField F E
β : E
hK : P K
⊢ P (restrictScalars F (↥K)⟮β⟯)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
|
cases' hK with α hK
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case intro
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
base : P ⊥
K : IntermediateField F E
β α : E
hK : F⟮α⟯ = K
⊢ P (restrictScalars F (↥K)⟮β⟯)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
|
rw [← hK, adjoin_simple_adjoin_simple]
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case intro
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
base : P ⊥
K : IntermediateField F E
β α : E
hK : F⟮α⟯ = K
⊢ P F⟮α, β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
|
haveI : Infinite F := isEmpty_fintype.mp F_inf
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case intro
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
base : P ⊥
K : IntermediateField F E
β α : E
hK : F⟮α⟯ = K
this : Infinite F
⊢ P F⟮α, β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
|
cases' primitive_element_inf_aux F α β with γ hγ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case intro.intro
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
base : P ⊥
K : IntermediateField F E
β α : E
hK : F⟮α⟯ = K
this : Infinite F
γ : E
hγ : F⟮α, β⟯ = F⟮γ⟯
⊢ P F⟮α, β⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
|
exact ⟨γ, hγ.symm⟩
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case inl
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_inf : IsEmpty (Fintype F)
P : IntermediateField F E → Prop := fun K => ∃ α, F⟮α⟯ = K
base : P ⊥
ih : ∀ (K : IntermediateField F E) (x : E), P K → P (restrictScalars F (↥K)⟮x⟯)
⊢ ∃ α, F⟮α⟯ = ⊤
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
|
exact induction_on_adjoin P base ih ⊤
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
|
Mathlib_FieldTheory_PrimitiveElement
|
case inr.intro
F : Type u_1
E : Type u_2
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsSeparable F E
F_finite : Fintype F
⊢ ∃ α, F⟮α⟯ = ⊤
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
·
|
exact exists_primitive_element_of_finite_bot F E
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
·
|
Mathlib.FieldTheory.PrimitiveElement.214_0.R5HND7n71i1v1rZ
|
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤
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Mathlib_FieldTheory_PrimitiveElement
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F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
|
wlog hmn : m < n
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case inr
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
this :
∀ (F : Type u_1) (E : Type u_2) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {α : E} {m n : ℕ},
m ≠ n → F⟮α ^ m⟯ = F⟮α ^ n⟯ → m < n → IsAlgebraic F α
hmn : ¬m < n
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
·
|
exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
·
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
|
by_cases hm : m = 0
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case pos
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : m = 0
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
·
|
rw [hm] at heq hmn
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
·
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case pos
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ 0⟯ = F⟮α ^ n⟯
hmn : 0 < n
hm : m = 0
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
|
simp only [pow_zero, adjoin_one] at heq
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case pos
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
hmn : 0 < n
hm : m = 0
heq : ⊥ = F⟮α ^ n⟯
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
|
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case pos.intro
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
hmn : 0 < n
hm : m = 0
heq : ⊥ = F⟮α ^ n⟯
y : F
h : (algebraMap F E) y = α ^ n
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
|
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case pos.intro
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
hmn : 0 < n
hm : m = 0
heq : ⊥ = F⟮α ^ n⟯
y : F
h : (algebraMap F E) y = α ^ n
⊢ (aeval α) (X ^ n - C y) = 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
|
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : ¬m = 0
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
|
replace hm : 0 < m := Nat.pos_of_ne_zero hm
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
|
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg.intro.intro
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m = (aeval (α ^ n)) r / (aeval (α ^ n)) s
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
|
by_cases hzero : aeval (α ^ n) s = 0
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case pos
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m = (aeval (α ^ n)) r / (aeval (α ^ n)) s
hzero : (aeval (α ^ n)) s = 0
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
·
|
simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
·
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case pos
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
hzero : (aeval (α ^ n)) s = 0
h : α = 0
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
|
exact h.symm ▸ isAlgebraic_zero
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m = (aeval (α ^ n)) r / (aeval (α ^ n)) s
hzero : ¬(aeval (α ^ n)) s = 0
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
·
|
rw [eq_div_iff hzero, ← sub_eq_zero] at h
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
·
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : ¬(aeval (α ^ n)) s = 0
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
|
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : ¬(aeval (α ^ n)) s = 0
⊢ s ≠ 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by
|
rintro rfl
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r : F[X]
h : α ^ m * (aeval (α ^ n)) 0 - (aeval (α ^ n)) r = 0
hzero : ¬(aeval (α ^ n)) 0 = 0
⊢ False
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl;
|
simp only [map_zero, not_true_eq_false] at hzero
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl;
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
|
let f : F[X] := X ^ m * expand F n s - expand F n r
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ IsAlgebraic F α
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
|
refine ⟨f, ?_, ?_⟩
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg.refine_1
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ f ≠ 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
·
|
have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
·
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ coeff f (n * natDegree s + m) ≠ 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
|
have hn : 0 < n := by linarith only [hm, hmn]
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ 0 < n
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by
|
linarith only [hm, hmn]
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
⊢ coeff f (n * natDegree s + m) ≠ 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
|
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
⊢ ¬n ∣ n * natDegree s + m
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
|
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
⊢ ¬n ∣ m
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
|
exact Nat.not_dvd_of_pos_of_lt hm hmn
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
hndvd : ¬n ∣ n * natDegree s + m
⊢ coeff f (n * natDegree s + m) ≠ 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
|
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
hndvd : ¬n ∣ n * natDegree s + m
⊢ leadingCoeff s ≠ 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
|
exact leadingCoeff_ne_zero.2 hzero
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg.refine_1
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
this : coeff f (n * natDegree s + m) ≠ 0
⊢ f ≠ 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
|
intro h
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg.refine_1
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h✝ : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
this : coeff f (n * natDegree s + m) ≠ 0
h : f = 0
⊢ False
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
|
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
case neg.refine_2
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ (aeval α) f = 0
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
·
|
simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
·
|
Mathlib.FieldTheory.PrimitiveElement.247_0.R5HND7n71i1v1rZ
|
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
⊢ FiniteDimensional F E
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
|
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
|
Mathlib.FieldTheory.PrimitiveElement.283_0.R5HND7n71i1v1rZ
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
IF : Type u_2 := { K // ∃ x, K = F⟮x⟯ }
⊢ FiniteDimensional F E
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
|
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
|
Mathlib.FieldTheory.PrimitiveElement.283_0.R5HND7n71i1v1rZ
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
IF : Type u_2 := { K // ∃ x, K = F⟮x⟯ }
this : ∀ (K : IF), FiniteDimensional F ↥↑K
⊢ FiniteDimensional F E
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
|
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
|
Mathlib.FieldTheory.PrimitiveElement.283_0.R5HND7n71i1v1rZ
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
IF : Type u_2 := { K // ∃ x, K = F⟮x⟯ }
this : ∀ (K : IF), FiniteDimensional F ↥↑K
hfin : FiniteDimensional F ↥(⨆ i, ↑i)
⊢ FiniteDimensional F E
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
|
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
|
Mathlib.FieldTheory.PrimitiveElement.283_0.R5HND7n71i1v1rZ
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
IF : Type u_2 := { K // ∃ x, K = F⟮x⟯ }
this : ∀ (K : IF), FiniteDimensional F ↥↑K
hfin : FiniteDimensional F ↥(⨆ i, ↑i)
htop : ⨆ K, ↑K = ⊤
⊢ FiniteDimensional F E
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
|
rw [htop] at hfin
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
|
Mathlib.FieldTheory.PrimitiveElement.283_0.R5HND7n71i1v1rZ
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
IF : Type u_2 := { K // ∃ x, K = F⟮x⟯ }
this : ∀ (K : IF), FiniteDimensional F ↥↑K
hfin : FiniteDimensional F ↥⊤
htop : ⨆ K, ↑K = ⊤
⊢ FiniteDimensional F E
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
|
exact topEquiv.toLinearEquiv.finiteDimensional
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
|
Mathlib.FieldTheory.PrimitiveElement.283_0.R5HND7n71i1v1rZ
|
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
⊢ ∃ α, F⟮α⟯ = K
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
exact topEquiv.toLinearEquiv.finiteDimensional
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
|
haveI := finiteDimensional_of_finite_intermediateField F E
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
|
Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
this : FiniteDimensional F E
⊢ ∃ α, F⟮α⟯ = K
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
exact topEquiv.toLinearEquiv.finiteDimensional
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
|
rcases finite_or_infinite F with (_ | _)
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
|
Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K
|
Mathlib_FieldTheory_PrimitiveElement
|
case inl
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
this : FiniteDimensional F E
h✝ : Finite F
⊢ ∃ α, F⟮α⟯ = K
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
exact topEquiv.toLinearEquiv.finiteDimensional
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
·
|
obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
·
|
Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K
|
Mathlib_FieldTheory_PrimitiveElement
|
case inl.intro
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
this : FiniteDimensional F E
h✝ : Finite F
α : ↥K
h : F⟮α⟯ = ⊤
⊢ ∃ α, F⟮α⟯ = K
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
exact topEquiv.toLinearEquiv.finiteDimensional
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
|
exact ⟨α, by simpa only [lift_adjoin_simple, lift_top] using congr_arg lift h⟩
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
|
Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K
|
Mathlib_FieldTheory_PrimitiveElement
|
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
this : FiniteDimensional F E
h✝ : Finite F
α : ↥K
h : F⟮α⟯ = ⊤
⊢ F⟮↑α⟯ = K
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
exact topEquiv.toLinearEquiv.finiteDimensional
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
exact ⟨α, by
|
simpa only [lift_adjoin_simple, lift_top] using congr_arg lift h
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
exact ⟨α, by
|
Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K
|
Mathlib_FieldTheory_PrimitiveElement
|
case inr
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
this : FiniteDimensional F E
h✝ : Infinite F
⊢ ∃ α, F⟮α⟯ = K
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
exact topEquiv.toLinearEquiv.finiteDimensional
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
exact ⟨α, by simpa only [lift_adjoin_simple, lift_top] using congr_arg lift h⟩
·
|
apply induction_on_adjoin (fun K ↦ ∃ α : E, F⟮α⟯ = K) ⟨0, adjoin_zero⟩
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
exact ⟨α, by simpa only [lift_adjoin_simple, lift_top] using congr_arg lift h⟩
·
|
Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K
|
Mathlib_FieldTheory_PrimitiveElement
|
case inr.ih
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
this : FiniteDimensional F E
h✝ : Infinite F
⊢ ∀ (K : IntermediateField F E) (x : E), (∃ α, F⟮α⟯ = K) → ∃ α, F⟮α⟯ = restrictScalars F (↥K)⟮x⟯
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
/-!
# Primitive Element Theorem
In this file we prove the primitive element theorem.
## Main results
- `exists_primitive_element`: a finite separable extension `E / F` has a primitive element, i.e.
there is an `α : E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`.
- `exists_primitive_element_iff_finite_intermediateField`: a finite extension `E / F` has a
primitive element if and only if there exist only finitely many intermediate fields between `E`
and `F`.
## Implementation notes
In declaration names, `primitive_element` abbreviates `adjoin_simple_eq_top`:
it stands for the statement `F⟮α⟯ = (⊤ : Subalgebra F E)`. We did not add an extra
declaration `IsPrimitiveElement F α := F⟮α⟯ = (⊤ : Subalgebra F E)` because this
requires more unfolding without much obvious benefit.
## Tags
primitive element, separable field extension, separable extension, intermediate field, adjoin,
exists_adjoin_simple_eq_top
-/
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField
namespace Field
section PrimitiveElementFinite
variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
/-! ### Primitive element theorem for finite fields -/
/-- **Primitive element theorem** assuming E is finite. -/
theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α⟯ = ⊤ := by
obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
apply eq_top_iff.mpr
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
simp only at hn
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self F (E := E) ↑α) n
#align field.exists_primitive_element_of_finite_top Field.exists_primitive_element_of_finite_top
/-- Primitive element theorem for finite dimensional extension of a finite field. -/
theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
∃ α : E, F⟮α⟯ = ⊤ :=
haveI : Finite E := finite_of_finite F E
exists_primitive_element_of_finite_top F E
#align field.exists_primitive_element_of_finite_bot Field.exists_primitive_element_of_finite_bot
end PrimitiveElementFinite
/-! ### Primitive element theorem for infinite fields -/
section PrimitiveElementInf
variable {F : Type*} [Field F] [Infinite F] {E : Type*} [Field E] (ϕ : F →+* E) (α β : E)
theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map] at hc
push_neg at hc
exact ⟨c, hc⟩
#align field.primitive_element_inf_aux_exists_c Field.primitive_element_inf_aux_exists_c
variable (F)
variable [Algebra F E]
/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
have hα := IsSeparable.isIntegral F α
have hβ := IsSeparable.isIntegral F β
let f := minpoly F α
let g := minpoly F β
let ιFE := algebraMap F E
let ιEE' := algebraMap E (SplittingField (g.map ιFE))
obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
let γ := α + c • β
suffices β_in_Fγ : β ∈ F⟮γ⟯
· use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
(C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
have h_ne_zero : h ≠ 0 :=
mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
· have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
-- Porting note: had to add `-map_add` to avoid going in the wrong direction.
simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
-map_add]
-- Porting note: an alternative solution is:
-- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
-- Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
-- Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
-- sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
rw [finale]
exact Subtype.mem (-p.coeff 0 / p.coeff 1)
have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
Algebra.smul_def, add_sub_cancel, minpoly.aeval]
· rw [eval_map, ← aeval_def, minpoly.aeval]
have h_splits : Splits ιEE' h :=
splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
intro x hx
rw [mem_roots_map h_ne_zero] at hx
specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
have f_root := root_left_of_root_gcd hx
rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
specialize hc x (by
rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
exact root_right_of_root_gcd hx)
by_contra a
apply hc
apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
ring
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]
trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])
· dsimp only
convert (gcd_map (algebraMap F⟮γ⟯ E)).symm
· simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, map_add, Polynomial.map_mul, map_X]
congr
#align field.primitive_element_inf_aux Field.primitive_element_inf_aux
-- If `F` is infinite and `E/F` has only finitely many intermediate fields, then for any
-- `α` and `β` in `E`, `F⟮α, β⟯` is generated by a single element.
-- Marked as private since it's a special case of
-- `exists_primitive_element_of_finite_intermediateField`.
private theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_simple_self F _
simp only [← heq] at αyβ_in_K
have β_in_K := sub_mem αxβ_in_K αyβ_in_K
rw [show (α + x • β) - (α + y • β) = (x - y) • β by rw [sub_smul]; abel1] at β_in_K
replace β_in_K := smul_mem _ β_in_K (x := (x - y)⁻¹)
rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K
have α_in_K : α ∈ F⟮α + x • β⟯ := by
convert ← sub_mem αxβ_in_K (smul_mem _ β_in_K)
apply add_sub_cancel
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
end PrimitiveElementInf
variable (F E : Type*) [Field F] [Field E]
variable [Algebra F E]
section SeparableAssumption
variable [FiniteDimensional F E] [IsSeparable F E]
/-- **Primitive element theorem**: a finite separable field extension `E` of `F` has a
primitive element, i.e. there is an `α ∈ E` such that `F⟮α⟯ = (⊤ : Subalgebra F E)`. -/
theorem exists_primitive_element : ∃ α : E, F⟮α⟯ = ⊤ := by
rcases isEmpty_or_nonempty (Fintype F) with (F_inf | ⟨⟨F_finite⟩⟩)
· let P : IntermediateField F E → Prop := fun K => ∃ α : E, F⟮α⟯ = K
have base : P ⊥ := ⟨0, adjoin_zero⟩
have ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮x⟯.restrictScalars F) := by
intro K β hK
cases' hK with α hK
rw [← hK, adjoin_simple_adjoin_simple]
haveI : Infinite F := isEmpty_fintype.mp F_inf
cases' primitive_element_inf_aux F α β with γ hγ
exact ⟨γ, hγ.symm⟩
exact induction_on_adjoin P base ih ⊤
· exact exists_primitive_element_of_finite_bot F E
#align field.exists_primitive_element Field.exists_primitive_element
/-- Alternative phrasing of primitive element theorem:
a finite separable field extension has a basis `1, α, α^2, ..., α^n`.
See also `exists_primitive_element`. -/
noncomputable def powerBasisOfFiniteOfSeparable : PowerBasis F E :=
let α := (exists_primitive_element F E).choose
let pb := adjoin.powerBasis (IsSeparable.isIntegral F α)
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F E).choose_spec
pb.map ((IntermediateField.equivOfEq e).trans IntermediateField.topEquiv)
#align field.power_basis_of_finite_of_separable Field.powerBasisOfFiniteOfSeparable
end SeparableAssumption
section FiniteIntermediateField
-- TODO: show a more generalized result: [F⟮α⟯ : F⟮α ^ m⟯] = m if m > 0 and α transcendental.
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
replace hm : 0 < m := Nat.pos_of_ne_zero hm
obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
by_cases hzero : aeval (α ^ n) s = 0
· simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
exact h.symm ▸ isAlgebraic_zero
· rw [eq_div_iff hzero, ← sub_eq_zero] at h
replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
let f : F[X] := X ^ m * expand F n s - expand F n r
refine ⟨f, ?_, ?_⟩
· have : f.coeff (n * s.natDegree + m) ≠ 0 := by
have hn : 0 < n := by linarith only [hm, hmn]
have hndvd : ¬ n ∣ n * s.natDegree + m := by
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
exact Nat.not_dvd_of_pos_of_lt hm hmn
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
coeff_expand hn r, hndvd, ite_false, sub_zero]
exact leadingCoeff_ne_zero.2 hzero
intro h
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
· simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem isAlgebraic_of_finite_intermediateField
[Finite (IntermediateField F E)] : Algebra.IsAlgebraic F E := fun α ↦
have ⟨_m, _n, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite fun n ↦ F⟮α ^ n⟯
isAlgebraic_of_adjoin_eq_adjoin F E hneq heq
theorem finiteDimensional_of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(isAlgebraic_of_finite_intermediateField F E x).isIntegral
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have htop : ⨆ K : IF, K.1 = ⊤ := le_top.antisymm fun x _ ↦
le_iSup (fun K : IF ↦ K.1) ⟨F⟮x⟯, x, rfl⟩ <| mem_adjoin_simple_self F x
rw [htop] at hfin
exact topEquiv.toLinearEquiv.finiteDimensional
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
exact ⟨α, by simpa only [lift_adjoin_simple, lift_top] using congr_arg lift h⟩
· apply induction_on_adjoin (fun K ↦ ∃ α : E, F⟮α⟯ = K) ⟨0, adjoin_zero⟩
|
rintro K β ⟨α, rfl⟩
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
exact ⟨α, by simpa only [lift_adjoin_simple, lift_top] using congr_arg lift h⟩
· apply induction_on_adjoin (fun K ↦ ∃ α : E, F⟮α⟯ = K) ⟨0, adjoin_zero⟩
|
Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ
|
theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K
|
Mathlib_FieldTheory_PrimitiveElement
|
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