Noether's problem for some 2-groups

TL;DR

This paper investigates Noether's problem for certain 2-groups, proving rationality of the fixed field under specific conditions related to group order, exponent, and field containing roots of unity.

Contribution

It establishes new rationality results for fixed fields of 2-groups with particular order and exponent constraints, extending previous knowledge in the area.

Findings

Proves rationality of $k(G)$ for groups of order $2^n$ with $n extgreater=4$ and exponent $2^e$ under specified conditions.

Shows that the fixed field $k(G)$ is rational when the field contains certain roots of unity.

Extends the class of groups for which Noether's problem has an affirmative answer.

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 $k$. We will prove that, 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.13A50,14E08,14M20,12F12