Noether's problem for the groups with a cyclic subgroup of index 4

TL;DR

This paper investigates when the fixed field of a finite group action on a rational function field is rational, providing new criteria for groups of order 2^n and 4n under certain conditions.

Contribution

It establishes new rationality results for Noether's problem for groups with a cyclic subgroup of index 4, extending previous knowledge.

Findings

Proves rationality for groups of order 2^n with specific exponent and field conditions.

Establishes rationality for groups of order 4n with certain element and field properties.

Identifies exceptional cases where rationality depends on the field's square classes.

Abstract

Let $G$ be a finite group and $k$ be a field. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\in G)^G$ is rational (i.e. purely transcendental) over $k$. Theorem 1. If $G$ is a group of order $2^n$ ($n\ge 4$) and of exponent $2^e$ such that (i) $e\ge n-2$ and (ii) $\zeta_{2^{e-1}} \in k$, then $k(G)$ is $k$-rational. Theorem 2. Let $G$ be a group of order $4n$ where $n$ is any positive integer (it is unnecessary to assume that $n$ is a power of 2). Assume that {\rm (i)} $\fn{char}k \ne 2$, $\zeta_n \in k$, and {\rm (ii)} $G$ contains an element of order $n$. Then $k(G)$ is rational over $k$, except for the case $n=2m$ and $G \simeq C_m \rtimes C_8$ where $m$ is an odd integer and the center of $G$ is of even order (note that $C_m$ is normal in $C_m \rtimes C_8$) ; for the exceptional case, $k(G)$ is rational over $k$ if and only if at least one of $-1, 2, -2$ belongs to $(k^{\times})^2$.