Noether's problems for 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 $\mathbf{C}(G)$ over $\mathbf{C}$ is non-trivial. Thus $\mathbf{C}(G)$ is not rational over $\mathbf{C}$. 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 ${\Phi }_7$.

arXiv admin note: text overlap with arXiv:1201.5555 by other authors

Cited by in corpus (2)

• Rationality problem for quasi-monomial actions
• Birational classification of fields of invariants for groups of order $128$