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from sympy.core.symbol import symbols
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.polys import QQ, ZZ
from sympy.polys.polytools import Poly
from sympy.polys.polyerrors import NotInvertible
from sympy.polys.agca.extensions import FiniteExtension
from sympy.polys.domainmatrix import DomainMatrix
from sympy.testing.pytest import raises
from sympy.abc import x, y, t
def test_FiniteExtension():
# Gaussian integers
A = FiniteExtension(Poly(x**2 + 1, x))
assert A.rank == 2
assert str(A) == 'ZZ[x]/(x**2 + 1)'
i = A.generator
assert i.parent() is A
assert i*i == A(-1)
raises(TypeError, lambda: i*())
assert A.basis == (A.one, i)
assert A(1) == A.one
assert i**2 == A(-1)
assert i**2 != -1 # no coercion
assert (2 + i)*(1 - i) == 3 - i
assert (1 + i)**8 == A(16)
assert A(1).inverse() == A(1)
raises(NotImplementedError, lambda: A(2).inverse())
# Finite field of order 27
F = FiniteExtension(Poly(x**3 - x + 1, x, modulus=3))
assert F.rank == 3
a = F.generator # also generates the cyclic group F - {0}
assert F.basis == (F(1), a, a**2)
assert a**27 == a
assert a**26 == F(1)
assert a**13 == F(-1)
assert a**9 == a + 1
assert a**3 == a - 1
assert a**6 == a**2 + a + 1
assert F(x**2 + x).inverse() == 1 - a
assert F(x + 2)**(-1) == F(x + 2).inverse()
assert a**19 * a**(-19) == F(1)
assert (a - 1) / (2*a**2 - 1) == a**2 + 1
assert (a - 1) // (2*a**2 - 1) == a**2 + 1
assert 2/(a**2 + 1) == a**2 - a + 1
assert (a**2 + 1)/2 == -a**2 - 1
raises(NotInvertible, lambda: F(0).inverse())
# Function field of an elliptic curve
K = FiniteExtension(Poly(t**2 - x**3 - x + 1, t, field=True))
assert K.rank == 2
assert str(K) == 'ZZ(x)[t]/(t**2 - x**3 - x + 1)'
y = K.generator
c = 1/(x**3 - x**2 + x - 1)
assert ((y + x)*(y - x)).inverse() == K(c)
assert (y + x)*(y - x)*c == K(1) # explicit inverse of y + x
def test_FiniteExtension_eq_hash():
# Test eq and hash
p1 = Poly(x**2 - 2, x, domain=ZZ)
p2 = Poly(x**2 - 2, x, domain=QQ)
K1 = FiniteExtension(p1)
K2 = FiniteExtension(p2)
assert K1 == FiniteExtension(Poly(x**2 - 2))
assert K2 != FiniteExtension(Poly(x**2 - 2))
assert len({K1, K2, FiniteExtension(p1)}) == 2
def test_FiniteExtension_mod():
# Test mod
K = FiniteExtension(Poly(x**3 + 1, x, domain=QQ))
xf = K(x)
assert (xf**2 - 1) % 1 == K.zero
assert 1 % (xf**2 - 1) == K.zero
assert (xf**2 - 1) / (xf - 1) == xf + 1
assert (xf**2 - 1) // (xf - 1) == xf + 1
assert (xf**2 - 1) % (xf - 1) == K.zero
raises(ZeroDivisionError, lambda: (xf**2 - 1) % 0)
raises(TypeError, lambda: xf % [])
raises(TypeError, lambda: [] % xf)
# Test mod over ring
K = FiniteExtension(Poly(x**3 + 1, x, domain=ZZ))
xf = K(x)
assert (xf**2 - 1) % 1 == K.zero
raises(NotImplementedError, lambda: (xf**2 - 1) % (xf - 1))
def test_FiniteExtension_from_sympy():
# Test to_sympy/from_sympy
K = FiniteExtension(Poly(x**3 + 1, x, domain=ZZ))
xf = K(x)
assert K.from_sympy(x) == xf
assert K.to_sympy(xf) == x
def test_FiniteExtension_set_domain():
KZ = FiniteExtension(Poly(x**2 + 1, x, domain='ZZ'))
KQ = FiniteExtension(Poly(x**2 + 1, x, domain='QQ'))
assert KZ.set_domain(QQ) == KQ
def test_FiniteExtension_exquo():
# Test exquo
K = FiniteExtension(Poly(x**4 + 1))
xf = K(x)
assert K.exquo(xf**2 - 1, xf - 1) == xf + 1
def test_FiniteExtension_convert():
# Test from_MonogenicFiniteExtension
K1 = FiniteExtension(Poly(x**2 + 1))
K2 = QQ[x]
x1, x2 = K1(x), K2(x)
assert K1.convert(x2) == x1
assert K2.convert(x1) == x2
K = FiniteExtension(Poly(x**2 - 1, domain=QQ))
assert K.convert_from(QQ(1, 2), QQ) == K.one/2
def test_FiniteExtension_division_ring():
# Test division in FiniteExtension over a ring
KQ = FiniteExtension(Poly(x**2 - 1, x, domain=QQ))
KZ = FiniteExtension(Poly(x**2 - 1, x, domain=ZZ))
KQt = FiniteExtension(Poly(x**2 - 1, x, domain=QQ[t]))
KQtf = FiniteExtension(Poly(x**2 - 1, x, domain=QQ.frac_field(t)))
assert KQ.is_Field is True
assert KZ.is_Field is False
assert KQt.is_Field is False
assert KQtf.is_Field is True
for K in KQ, KZ, KQt, KQtf:
xK = K.convert(x)
assert xK / K.one == xK
assert xK // K.one == xK
assert xK % K.one == K.zero
raises(ZeroDivisionError, lambda: xK / K.zero)
raises(ZeroDivisionError, lambda: xK // K.zero)
raises(ZeroDivisionError, lambda: xK % K.zero)
if K.is_Field:
assert xK / xK == K.one
assert xK // xK == K.one
assert xK % xK == K.zero
else:
raises(NotImplementedError, lambda: xK / xK)
raises(NotImplementedError, lambda: xK // xK)
raises(NotImplementedError, lambda: xK % xK)
def test_FiniteExtension_Poly():
K = FiniteExtension(Poly(x**2 - 2))
p = Poly(x, y, domain=K)
assert p.domain == K
assert p.as_expr() == x
assert (p**2).as_expr() == 2
K = FiniteExtension(Poly(x**2 - 2, x, domain=QQ))
K2 = FiniteExtension(Poly(t**2 - 2, t, domain=K))
assert str(K2) == 'QQ[x]/(x**2 - 2)[t]/(t**2 - 2)'
eK = K2.convert(x + t)
assert K2.to_sympy(eK) == x + t
assert K2.to_sympy(eK ** 2) == 4 + 2*x*t
p = Poly(x + t, y, domain=K2)
assert p**2 == Poly(4 + 2*x*t, y, domain=K2)
def test_FiniteExtension_sincos_jacobian():
# Use FiniteExtensino to compute the Jacobian of a matrix involving sin
# and cos of different symbols.
r, p, t = symbols('rho, phi, theta')
elements = [
[sin(p)*cos(t), r*cos(p)*cos(t), -r*sin(p)*sin(t)],
[sin(p)*sin(t), r*cos(p)*sin(t), r*sin(p)*cos(t)],
[ cos(p), -r*sin(p), 0],
]
def make_extension(K):
K = FiniteExtension(Poly(sin(p)**2+cos(p)**2-1, sin(p), domain=K[cos(p)]))
K = FiniteExtension(Poly(sin(t)**2+cos(t)**2-1, sin(t), domain=K[cos(t)]))
return K
Ksc1 = make_extension(ZZ[r])
Ksc2 = make_extension(ZZ)[r]
for K in [Ksc1, Ksc2]:
elements_K = [[K.convert(e) for e in row] for row in elements]
J = DomainMatrix(elements_K, (3, 3), K)
det = J.charpoly()[-1] * (-K.one)**3
assert det == K.convert(r**2*sin(p))
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