Noether's problem for p-groups with three generators

TL;DR

This paper proves that for certain nonabelian p-groups with three generators, the fixed field under the group action is rational over the base field, extending known results to groups of order p^5.

Contribution

It establishes the rationality of the fixed field for a class of nonabelian p-groups with three generators, generalizing Noether's problem results.

Findings

Fixed field is rational over the base field for specified p-groups.

Rationality holds when the base field contains a primitive p^a-th root of unity.

Application to groups of order p^5 with three generators.

Abstract

Let $p$ be an odd prime and $G$ be a nonabelian group of order $p^{n}$ with the presentation $$<\alpha,\beta,\gamma\mid \alpha^{p^{a}}=\beta^{p^{b}}=\gamma^{p^{c}}=1, [\alpha,\gamma]=1,[\gamma,\beta]=\alpha^{p^{r}},[\alpha,\beta]=\gamma^{p^{e}}>,$$ where $n>a\geq b\geq c\geq 1$. Let $k$ be a field containing a primitive $p^{a}$-th root of unity and $G$ act on the rational function field $k(x_{h}:h\in G)$ by $g\cdot x_{h}=x_{gh}$ for all $g,h\in G$. In this note, we prove that the fixed field $k(G)=k(x_{h}:h\in G)^{G}$ is rational over $k$. As a corollary, we prove that if $k$ contains a primitive $p^{4}$-th root of unity and $G$ is a nonabelian group of order $p^{5}$ generated by three elements, then $k(G)$ is rational over $k$.