Central extensions of \(S_ n\) as Galois groups via trinomials
The main result states that every central extension of the symmetric group \(S_ n\) is a Galois group over every number field K, for \(n\equiv 0,1 (mod 8)\), \(n\equiv 2 (mod 8)\) and n sum of two squares, or \(n\equiv 3\) (mod 8) and n-1 a sum of two squares; also, if \(\sqrt{2}\not\in K\), for \(n\equiv 2\) (mod 8) (or \(n\equiv 3\) (mod 8)) and n (or n-1) of the form \(K^ 2+2Y^ 2\). A reduction lemma [cf. \textit{D. Kotlar, M. Schacher} and the author, J. Algebra 124, No.1, 183-198 (1989; Zbl 0679.12011)] shows that it is enough to prove that \(S_ n^{\pm}\times_{C_ 2}C_{2^ m}\) is a Galois group over K, for every m, where \(S_ n^{\pm}\) are the double covers of \(S_ n\). In order to prove this, the author considers trinomials \(X^ n+aX+b\) with Galois group \(S_ n\), given discriminant and such that the embedding problem associated to \(S^+_ n\) or to \(S^-_ n\) has a solution. Remark a mistake in theorem 1: In fact, with the author's notations, \(S^-_ n\) and \(S^+_ n\) must be interchanged in the assertions of theorem 1.
scientific article; zbMATH DE number 4142156