Noether's problem for abelian extensions of cyclic $p$-groups

Ivo M. Michailov

Vol. 270 (2014), No. 1, 167–189

DOI: 10.2140/pjm.2014.270.167

Abstract

Let $K$ be a field and $G$ a finite group. Let $G$ act on the rational function field $K\left(x\left(g\right):g\in G\right)$ by $K$-automorphisms defined by $g\cdot x\left(h\right)=x\left(gh\right)$ for any $g$, $h\in G$. Denote by $K\left(G\right)$ the fixed field $K{\left(x\left(g\right):g\in G\right)}^G$. Noether’s problem then asks whether $K\left(G\right)$ is rational (i.e., purely transcendental) over $K$. The first main result of this article is that $K\left(G\right)$ is rational over $K$ for a certain class of $p$-groups having an abelian subgroup of index $p$. The second main result is that $K\left(G\right)$ is rational over $K$ for any group of order $p^5$ or $p^6$ (where $p$ is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if $charK\ne p$ then $K$ contains a primitive $p^e$-th root of unity, where $p^e$ is the exponent of $G$.)

In loving memory of my dear mother

Keywords

Noether's problem, rationality problem, metabelian group actions

Mathematical Subject Classification 2010

Primary: 14E08, 14M20

Secondary: 13A50, 12F12

Milestones

Received: 18 January 2013

Accepted: 13 February 2013

Published: 2 August 2014