On stem extensions of \(S_ n\) as Galois group over number fields
type: Item
instance of: scholarly article
title: On stem extensions of \(S_ n\) as Galois group over number fields
zbMATH Open document ID: 0662.12011
DOI: 10.1016/0021-8693(88)90205-0
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Many years ago Emmy Noether asked whether every finite group can be realized as a Galois group over the rational number field. The present paper deals with the problem of the realizability of the finite stem extensions of the symmetric group \(S_ n\) as a Galois group over number fields. A stem extension of a finite group G is a central extension \(1\to A\to E\to G\to 1\), in which \(A\subset E'\) (the commutator group of E). The author gives a wide class of polynomials with such Galois groups. For example, if \(n\equiv 0,1,2,\quad or\quad 3\quad (mod 8),\) then every finite stem extension of \(S_ n\) occurs as a Galois group over every number field K. Moreover, if \(i\in K\), then every finite stem extension of \(S_ n\) can be realized as a Galois group over K for every value of n.
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 4081686
zbMATH Keywords: inverse problem of Galois theory
zbMATH Keywords: stem extension of a finite group
author: Nuria Vila
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90205-0
OpenAlex ID: W2094708217
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description: scientific article; zbMATH DE number 4081686