Double covers of the symmetric groups as Galois groups over number fields
The main result of the paper states that for \(n\geq 4\) both double covers \(S^{+}_n\) and \(S^{-}_n\) of the symmetric group \(S_n\) are realizable as Galois group over every number field provided that \(n\equiv 0,1,2,\) or \(3\pmod 8\). If \(n\equiv 4\) or \(5\pmod 8\), or if \(n\equiv 6\) or \(7\pmod 8\), then still \(S^{-}_n\), respectively \(S^{+}_n\), is always realizable. It is also shown that the trinomial \(x^n+ntx+(n-1)t\) can be specialized in order to serve for the above result. A main ingredient of the proof is a formula due to Serre which relates the obstruction to the embedding problem \(S^{-}_n\to S_n=\mathrm{Gal}(K/k)\) to the Witt invariant of a trace form.
scientific article; zbMATH DE number 4089667
scientific article