Noether's problems for groups of order 243

TL;DR

This paper investigates Noether's problem for groups of order 243, establishing conditions under which the fixed field is rational over the base field, with specific results depending on the group's isoclinism family.

Contribution

It classifies when the fixed field is rational for groups of order 243, extending understanding of Noether's problem in this specific case.

Findings

For groups of order 243, the fixed field is rational over k if ζ_9 ∈ k, except possibly for certain isoclinism families.

The paper identifies three groups with non-trivial unramified Brauer group, indicating non-rationality.

Provides conditions on the base field k for rationality of the fixed field for most groups of order 243.

Abstract

Let $k$ be any field, $G$ be a finite group. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $h\cdot x_g=x_{hg}$ for any $g,h\in G$. Denote by $k(G)=k(x_g:g\in G)^G$ the fixed field. Noether's problem asks, under what situations, the fixed field $k(G)$ will be rational (= purely transcendental) over $k$. According to the data base of GAP there are $10$ isoclinism families for groups of order $243$. It is known that there are precisely $3$ groups $G$ of order $243$ (they consist of the isoclinism family $\Phi_{10}$) such that the unramified Brauer group of $\bm{C}(G)$ over $\bm{C}$ is non-trivial. Thus $\bm{C}(G)$ is not rational over $\bm{C}$. We will prove that, if $\zeta_9 \in k$, then $k(G)$ is rational over $k$ for groups of order $243$ other than these $3$ groups, except possibly for groups belonging to the isoclinism family $\Phi_7$.