Double covers of the symmetric groups as Galois groups over number fields
type: Item
instance of: scholarly article
title: Double covers of the symmetric groups as Galois groups over number fields
zbMATH Open document ID: 0666.12004
DOI: 10.1016/0021-8693(88)90204-9
author: Jack Sonn
author: Murray M. Schacher
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F12
zbMATH DE Number: 4089667
zbMATH Keywords: inverse problem of Galois theory
zbMATH Keywords: double covers of the symmetric group
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90204-9
OpenAlex ID: W2029126737
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description: scientific article; zbMATH DE number 4089667