Double covers of \(S_5\) and Frobenius groups as Galois groups over number fields (Q1104976)

In this article, it is first proved that the double covers \(S^+_5\) and \(S^-_5\) of the symmetric group \(S_5\) are realizable as Galois groups over any number field. Recently, Feit has shown that the same holds for the double cover \(A^+_5\cong \mathrm{SL}(2,5)\) of \(A_5\), by computing the Witt invariant of generalized Laguerre polynomials. A previous result of the author himself in [Isr. J. Math. 31, 91--96 (1978; Zbl 0391.12004)] (the reference of which in the paper under review is [11,2.7] and not [1,2.7] as indicated in the introduction) yields then to the following theorem: Every Frobenius group is realizable as a Galois group over every number field. The method of proof is to use Serre's formula about the Witt invariant of the trace form [\textit{J.-P. Serre}, Comment. Math. Helv. 59, 651--676 Zbl 0565.12014)], in conjunction with results of Feit, produce infinitely many linearly disjoint \(S^+_5\) or \(S^-_5\) extensions of \(\mathbb{Q}\).