半直積群上的Noether問題

Abstract

令K為一體，G為有限群。 定義群$G$作用於(acts on)有理函數體L = K(x_{sigma} : sigma in G)上， 此處 au cdot x_{sigma} = x_{ ausigma}, forall au, sigma in G。 令K(G) = L^{G} = { frac{f}{g} in L : sigma(frac{f}{g}) = frac{f}{g}, forall sigma in G } 為此作用(action)的固定體(fixed field)。 Noether問題就是要決定K(G)在K之上是否為有理(rational)的(=purely transcendental, 純超越的。) 考慮兩循環群(cyclic group)C_m, C_n的半直積群G = C_m times C_n. 目前我們已知若mathbb{Z}[zeta_n]為唯一分解整環(unique factorization domain)， 且K包含足夠的單位根，則K(G)是有理的。 但尚未有人給出一對質數p, q的反例，使得mathbb{C}(C_p times C_q)為非有理的。 本文給出K(C_m times C_n)為有理的必要條件。

Let $K$ be a field, $G$ a finite group. Let $G$ act on function field $L = K(x_{sigma} : sigma in G)$ by $ au cdot x_{sigma} = x_{ ausigma}$ for any $sigma, au in G$. Denote the fixed field of the action by $K(G) = L^{G} = { frac{f}{g} in L : sigma(frac{f}{g}) = frac{f}{g}, forall sigma in G }$. Noether's problem asks whether $K(G)$ is rational (purely transcendental) over $K$. It is known that if $G = C_m times C_n is semidirect product of cyclic groups C_m, C_n with mathbb{Z}[zeta_n] a unique factorization domain, and K contains an eth primitive root of unity, where e is the exponent of G. Then K(G) is rational over K. But it is still an open question whether there exists prime pair p, q such that mathbb{C}(C_p times C_q) is not rational over mathbb{C}. In this paper, we show that, under some conditions, K(C_m times C_n) is rational over K.