Subgroups of \(\text{Hol} Q_ 8\) as Galois groups
type: Item
instance of: scholarly article
title: Subgroups of \(\text{Hol} Q_ 8\) as Galois groups
zbMATH Open document ID: 0849.12007
author: Arne Ledet
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: Let \(p\) be a prime and let \(L/K\) be a Galois extension of fields of characteristic \(\neq p\). The main theorem of the paper under review states that the solvability of the embedding problem given by the non-split group extension \[ 1 \to C_p \to E \to G(L/K) \to 1 \] with kernel of order \(p\) is equivalent to the solvability of a central embedding problem over a field \(K' \supset K\) which contains the \(p\)-th roots of unity. The reduction is effective. All solutions of the first embedding problem can be obtained from a single solution of the latter embedding problem. Finally the author applies his reduction to compute the obstruction to certain embedding problems, where \(E\) is one of four subgroups of the holomorph \(\text{Hol}(Q_8) \simeq Q_8 \rtimes \Aut (Q_8) \simeq Q_8 \rtimes S_4\) of the quaternion group \(Q_8\) including the holomorph itself.
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 883770
zbMATH Keywords: Galois extension
zbMATH Keywords: embedding problem
zbMATH Keywords: obstruction
zbMATH Keywords: holomorph
zbMATH Keywords: quaternion group
reviewed by: Martin Epkenhans
MaRDI profile type: Publication
full work available at URL: jabr.1996.0130
OpenAlex ID: W1970215742
DOI: 10.1006/JABR.1996.0130
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description: scientific article; zbMATH DE number 883770