Noether's problem for some 2-groups

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

The content of this paper became part of a new paper "Noether's problem for the groups with a cyclic subgroup of index 4" which has been posted in arXiv