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

TL;DR

This paper proves the rationality of fixed fields under certain p-group actions, specifically for groups of nilpotency class 2 with the ABC property and some 3-generator p-groups, advancing understanding of Noether's problem.

Contribution

It establishes the rationality of $K(G)$ for p-groups of nilpotency class 2 with the ABC property and for specific 3-generator p-groups, extending known cases of Noether's problem.

Findings

Proves $K(G)$ is rational for p-groups of nilpotency class 2 with ABC property.

Establishes rationality for two specific 3-generator p-groups.

Addresses cases under different characteristic and root of unity assumptions.

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 $K$. Let $p$ be any prime and let $G$ be a $p$-group of exponent $p^e$. Assume also that {\rm (i)} char $K = p>0$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. In this paper we prove that if $G$ is any $p$-group of nilpotency class 2, which has the ABC (Abelian-By-Cyclic) property, then $K(G)$ is rational over $K$. We also prove the rationality of $K(G)$ over $K$ for two 3-generator $p$-groups $G$ of arbitrary nilpotency class.