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- W2463446470 abstract "Let $k$ be any field, $G$ be a finite group. Let $G$ act on the rational function field $k(x_g:gin G)$ by $k$-automorphisms defined by $hcdot x_g=x_{hg}$ for any $g,hin G$. Denote by $k(G)=k(x_g:gin G)^G$ 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 $Phi_7$." @default.
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- W2463446470 date "2014-03-03" @default.
- W2463446470 modified "2023-09-27" @default.
- W2463446470 title "Noether's problems for groups of order 243" @default.
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