Noether's problem for \hat{S}_4 and \hat{S}_5

Abstract

Let $k$ be a field, $G$ be a finite group and $k(x_g:g\in G)$ be the rational function field over $k$, on which $G$ acts by $k$-automorphisms defined by $h\cdot x_g=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the fixed subfield $k(G):=k(x_g:g\in G)^G$ is $k$-rational, i.e.\ purely transcendental over $k$. If $\widehat{S_n}$ is the double cover of the symmetric group $S_n$, in which the liftings of transpositions and products of disjoint transpositions are of order $4$, Serre shows that $\bm{Q}(\widehat{S_4})$ and $\bm{Q}(\widehat{S_5})$ are not $\bm{Q}$-rational. We will prove that, if $k$ is a field such that $\fn{char} k \neq 2, 3$, and $k(\zeta_8)$ is a cyclic extension of $k$, then $k(\widehat{S_4})$ is $k$-rational. If it is assumed furthermore that $\fn{char}k=0$, then $k(\widehat{S_5})$ is also $k$-rational.