Central extensions of \(S_ n\) as Galois groups via trinomials
type: Item
instance of: scholarly article
title: Central extensions of \(S_ n\) as Galois groups via trinomials
zbMATH Open document ID: 0697.12008
DOI: 10.1016/0021-8693(89)90168-3
author: Jack Sonn
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: 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.
reviewed by: Nuria Vila
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 4142156
zbMATH Keywords: central extension of symmetric group
zbMATH Keywords: trinomials
zbMATH Keywords: embedding problem
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90168-3
OpenAlex ID: W2025452104
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description: scientific article; zbMATH DE number 4142156