Galois realizability of groups of order 64
type: Item
instance of: scholarly article
title: Galois realizability of groups of order 64
zbMATH Open document ID: 1256.12003
author: Tara L. Smith
author: H. G. Grundman
published in: Central European Journal of Mathematics
publication date: 2011-05-27T00:00:00Z
review text: Given a field \(K\) and a group \(G\), the inverse problem of Galois theory asks whether or not there exists a Galois field extension \(L\) of \(K\) such that the Galois group of \(L\) over \(K\) is isomorphic to \(G\). In the article under review the authors determine necessary and sufficient conditions for the realizability of 134 non abelian groups of order 64 as Galois groups. The conditions are given in terms of triviality of specific elements (called \textit{obstructions}) in the Brauer group of \(K\). The obstructions are expressed as products of quaternion algebras and are written in several tables. The authors provide two sample proofs of the main types of embedding problems related to the groups. The first type produces the obstructions for 34 groups with a quotient of the type \((C_2)^r\times (C_4)^s\times (D_8)^t\). The second type produces 100 obstructions for groups known as \textit{pullbacks} (i.e., having more than one central element of order 2).
reviewed by: Ivo M. Michailov
Mathematics Subject Classification ID: 12F12
zbMATH DE Number: 5900614
zbMATH Keywords: inverse Galois theory
zbMATH Keywords: 2-groups
zbMATH Keywords: obstructions
zbMATH Keywords: embedding problem
zbMATH Keywords: quaternion algebra
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s11533-010-0052-1
OpenAlex ID: W1970332863
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DOI: 10.2478/S11533-010-0052-1
description: scientific article; zbMATH DE number 5900614