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

TL;DR

This paper investigates conditions under which the fixed field of a group action on a rational function field is purely transcendental, focusing on abelian extensions of cyclic p-groups and specific p-groups of order p^5 and p^6.

Contribution

It proves rationality of the fixed field for certain p-groups with abelian subgroups or normal subgroups, extending known cases of Noether's problem.

Findings

K(G) is rational for p-groups with an abelian subgroup of index p

K(G) is rational for groups of order p^5 or p^6 with an abelian normal subgroup and cyclic quotient

Results depend on the field containing primitive roots of unity when char K ≠ p

Abstract

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