Noether's problem for \(\hat{S}_4\) and \(\hat{S}_5\) (Q453228)

Let \(k\) be a field and \(G\) be a finite group. Let \(G\) act on the rational function field \(k(x(g):g\in G)\) by \(k\) automorphisms defined by \(g\cdot x(h)=x(gh)\) for any \(g,h\in G\). Denote by \(k(G)\) the fixed field \(k(x(g):g\in G)^G\). Noether's problem then asks whether \(k(G)\) is rational (i.e., purely transcendental) over \(k\). In the paper under review, the authors investigate the Noether's problem for \(G=\hat S_4\) and \(G=\hat S_5\), where by \(\hat S_n\) is denoted the double cover of the symmetric group \(S_n\), in which the liftings of transpositions and products of disjoint transpositions are of order \(4\).NEWLINENEWLINESerre proved in the book [\textit{S. Garibaldi, A. Merkurjev} and \textit{J.-P. Serre}, Cohomological invariants in Galois cohomology. University Lecture Series 28. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1159.12311)] that \(\mathbb Q(\hat S_4)\) and \(\mathbb Q(\hat S_5)\) are not \(\mathbb Q\)-rational.NEWLINENEWLINEThe main result of Kang and Zhou's paper is that if \(k\) is a field such that \(\mathrm{char} k\neq 2,3\) and \(k(\zeta_8)\) is a cyclic extension of \(k\), then \(k(\hat S_4)\) is \(k\)-rational. If it is assumed furthermore that \(\mathrm{char} k=0\), then \(k(\hat S_5)\) is also \(k\)-rational. The main idea of the proof is to use the method of Galois descent, namely first is solved the rationality of \(k(\zeta_8)(\hat S_4)\), and then descend the ground field to \(k\). The authors also construct a \(4\)-dimensional faithful representation of \(\hat S_4\) defined over the field \(k\). This representation allows to decrease the number of variables by applying some well known rationality criteria.