Abstract
Abstract. Let K be any field and G be a finite group. Let G act on the rational function field
Abstract
Noether's problem asks whether K(G) is rational (= purely transcendental) over K. We shall prove that K(G) is rational over K if G is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.
§1. Introduction
Let K be any field and G be a finite group. Let G act on the rational function field K(x g : g ∈ G) by K-automorphisms such that g · x h = x gh for any g, h ∈ G. Denote by K(G) the fixed field K(x g : g ∈ G) G = {f ∈ K(x g : g ∈ G) : σ · f = f for any σ ∈ G}. Noether's problem asks whether K(G) is rational (=purely transcendental) over K.
§1. Introduction
Noether's problem is related to the inverse Galois problem, which asks whether there is a Galois extension L over K such that Gal(L/K) G if the field K and the finite group G are prescribed. In fact, if K is an infinite field and K(G) is rational over K, then there exists a generic Galois G-extension over the field K [Sa1, Theorem 5.1]; see Proposition 2.2 for a generalization. A generic Galois G-extension is some universal object of G-extensions such that we can apply Hilbert irreducibility theorem; see [Sa1; Me] for more details. When K is a Hilbertian field, i.e. Hilbert irreducibility theorem is valid for irreducible polynomials f ∈ K[x 1 , · · · , x n ], the existence of a generic Galois G-extension over K will guarantee the existence of Notations and terminologies. A field extension L over K is rational if L is purely transcendental over K; L is called stably rational over K if there exist elements y 1 , · · · , y N which are algebraically independent over L such that L(y 1 , · · · , y N ) is rational over K. The dihedral group of order 2n is defined as σ, τ : σ n = τ 2 = 1, τστ −1 = σ −1 , which is denoted by D n . The quaternion group of order 8 is defined as σ, τ :
§1. Introduction
G . The representation space of the regular representation of G over K is denoted by
§1. Introduction
We recall a variant of Hilbertsatz 90 which has been used by many people under different disguises.
Theorem 2.1 ([HK2, Theorem 1]). Let L be a field and G be a finite group acting on
where Proof. Let W be the representation space of the regular representation of G. Thus
Theorem 2.1 ([HK2, Theorem 1]). Let L be a field and G be a finite group acting on
G is also stably rational. Hence there exists a positive integer m such that
Theorem 2.1 ([HK2, Theorem 1]). Let L be a field and G be a finite group acting on
All we need to do is to define a G-equivariant map ϕ :
Theorem 2.1 ([HK2, Theorem 1]). Let L be a field and G be a finite group acting on
s is a unit. Note that we should prove a "multi-variable" version of [Sa1, Lemma 5.2] . But this is not difficult and is omitted.
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
Proposition 2.6. Let char K = 0 or char K n and let ζ be a primitive n-th root Vol. 79 (2004) Noether's problem for dihedral 2-groups 151
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
Remark. Compare with [Bl, Proposition 2.1 and Theorem 3.3] .
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
Dn is rational over K.
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
Apply Theorem 2.1. We get
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
Note that 
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
<σ,τ,ρ> is rational over K, so is K (ζ)(t, y 1 ) <σ,τ,ρ> over K. Define m = n, if n is an odd integer; m = n/2 if n is an even integer. Then the restriction of σ to K(ζ)(t) is of order m.
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
Theorem 2.7 (Gröbner [Gr] ). Let G be the quaternion group. Then K(G) is rational over K for any field K.
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
Proof. Because of Theorem 2.5, we may assume that char K = 2. Recall the notations at the end of Section 1. We write G = σ, τ :
Theorem 2.4 ([AHK, Theorem 3.1]). Let G be any group whose order may be finite or infinite. Suppose that G acts on L(x), the rational function field of one variable over a field L. Assume that, for any
Note that V is a faithful subrepresentation of the regular representation W = ⊕ g∈G K ·x(g). In fact, we may take
Now apply Theorem 2.1. Thus it remains to prove that
It is straightforward to check that Define z 1 = y 1 , z 2 = y 2 /y 1 , z 3 = y 3 . Then
Now apply Theorem 2.1. Thus it remains to prove that
<σ> . Now apply Theorem 2.3 with
Vol. 79 (2004)
Noether's problem for dihedral 2-groups 153
Vol. 79 (2004)
Now it is easy to check that
Vol. 79 (2004)
by the last statement of Theorem 2.3 (here a = b = −1 in the present situation), we find that
Vol. 79 (2004)
where A = −(x 2 + 1)(x 2 − p 2 ). Define t = p 2 , q 1 = y + (A/y), q 2 = p{y − (A/y)}. We find that
Vol. 79 (2004)
with the relation q (p, x, y) . (Note that the last equality holds because we can solve y within the field K(x, q 1 , y − (A/y), p).) Now we will simplify the relation (1). It becomes
Vol. 79 (2004)
Dividing by t 2 on both sides, we get
Vol. 79 (2004)
From (2), it is obvious that t ∈ K(x, q 1 , q 2 /t). Thus
Vol. 79 (2004)
is rational over K. 
Main results
Without loss of generality we will assume that K is any field with charK = 2 throughout this section, because Theorem 2.5 will take care of the case char K = 2.
Main results
, which is the definition of this group given in [Le1] .
Main results
We will find a faithful subrepresentation of W = ⊕ g∈G K · x(g), the regular representation of G. Define
Main results
where the index i + 1 or ai is understood to be taken modulo 8. By Theorem 2.1, in order to prove the rationality problem of K(G), it suffices to consider the case of
Main results
In the sequel, we shall take the following convention: if we write the action of ρ, it is understood that √ −1 / ∈ K; however, if √ −1 ∈ K, the reader can just forget ρ even when we write the action of it. We will extend the actions of σ, τ, ρ to K(
Main results
Define
Main results
Vol. 79 (2004) Noether's problem for dihedral 2-groups 155
Main results
and the actions of σ, τ, ρ are given by
Main results
Thus the action of στ is given by
Main results
By Theorem 2.3, define
Main results
where a = √ −1(t + 2), b = t 2 − 4, and we find that (s, t, u, v) . (s, t, u, v) = K(s, t, u, v) is rational as we expect. From now on, assume √ −1 / ∈ K and ρ actually exists. We shall find the action of ρ on u and v. Remember ρ( 
Main results
Here is a cheating way to demonstrate the above identity. By Theorem 2.3, the right-hand side of Identity (3) is equal to (y + (b/y) − (1/u)) −1 . It is really routine to check that the left-hand side of Identity (3) is equal to (y + (b/y) − (1/u)) −1 . In conclusion, we find ρ(u) = c/u where c = w 2 /{(t 2 − 4)w 2 + 2(t + 2)}. Define p = λu/w where λ = (t 2 − 4)w 2 + 2(t + 2). Then ρ(p) = λ/p. Now K( √ −1) (s, t, u, v) is rational over K.
Theorem 3.2. If G is the quasi-dihedral group, then K(G) is rational over K.
Proof. We shall show that K(x 0 , · · · , x 7 ) G is rational over K where σ :
Theorem 3.2. If G is the quasi-dihedral group, then K(G) is rational over K.
The proof is almost the same as that in Theorem 3.1. We shall make the same change of variables, but the action may not be the same as in the proof of Theorem 3.1. We shall indicate the main modifications.

{'paper_id': '121245533', 'title': "Noether's problem for dihedral 2-groups II", 's2_url': 'https://api.semanticscholar.org/CorpusID:121245533', 'source': 'S2ORC'}