Generic Galois extensions for SL\(_2(\mathbb F_5)\) over \(\mathbb Q\) (Q2459330)

In inverse Galois theory it is an important question to decide for which finite groups \(G\) there exists a \(G\)-generic polynomial over the field \(k\), in particular for the case \(k=\mathbb{Q}\). If the base field \(k\) is infinite the existence of a \(G\)-generic polynomial over \(k\) is equivalent to the existence of a \(G\)-generic extension of \(k\). The present paper primarily deals with the case where \(G\) is the non-trivial double cover \(\widetilde A_n\) of the alternating group \(A_n\). It has been proved by J.-F. Mestre and Y. Rikuna that there exists a generic \(\widetilde A_4\)-extension over \(\mathbb{Q}\). \textit{J.-P. Serre} [Cohomological invariants, Witt invariants, and trace forms, Univ. Lecture Ser. 28, 1--100, Am. Math. Soc., Providence, RI (2003; Zbl 1159.12311)] has proved that there is no generic \(\widetilde A_n\)-extension for \(n= 6\) or \(n= 7\). In the paper under review the following interesting result is obtained: Let \(k\) be a field of characteristic 0 and \(n\) an odd integer \(\geq 5\). Then there exists a generic \(\widetilde A_n\)-extension over \(k\) if and only if there exists a generic \(\widetilde{A_{n-1}}\)-extension over \(k\). In view of the results by J.-F. Mestre and Y. Rikuna it follows that there exists a generic \(\widetilde A_5(=\text{SL}_2(\mathbb{F}_5))\)-extension over \(\mathbb{Q}\).