Double covers of \(S_5\) and Frobenius groups as Galois groups over number fields
type: Item
instance of: scholarly article
title: Double covers of \(S_5\) and Frobenius groups as Galois groups over number fields
zbMATH Open document ID: 0648.12006
DOI: 10.1016/0021-8693(88)90301-8
author: Jack Sonn
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: In this article, it is first proved that the double covers \(S^+_5\) and \(S^-_5\) of the symmetric group \(S_5\) are realizable as Galois groups over any number field. Recently, Feit has shown that the same holds for the double cover \(A^+_5\cong \mathrm{SL}(2,5)\) of \(A_5\), by computing the Witt invariant of generalized Laguerre polynomials. A previous result of the author himself in [Isr. J. Math. 31, 91--96 (1978; Zbl 0391.12004)] (the reference of which in the paper under review is [11,2.7] and not [1,2.7] as indicated in the introduction) yields then to the following theorem: Every Frobenius group is realizable as a Galois group over every number field. The method of proof is to use Serre's formula about the Witt invariant of the trace form [\textit{J.-P. Serre}, Comment. Math. Helv. 59, 651--676 Zbl 0565.12014)], in conjunction with results of Feit, produce infinitely many linearly disjoint \(S^+_5\) or \(S^-_5\) extensions of \(\mathbb{Q}\).
reviewed by: Richard Massy
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 4057629
zbMATH Keywords: symmetric group \(S_5\)
zbMATH Keywords: Frobenius group realizable as Galois group
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90301-8
OpenAlex ID: W2129314823
cites work: \({\mathbb{Q}}\)-admissibility of certain nonsolvable groups
cites work: Sylow-metacyclic groups and Q-admissibility
cites work: Examples of some \({\mathbb{Q}}\)-admissible groups
cites work: Cohomological topics in group theory.
cites work: Zum Einbettungsproblem.
cites work: Q4061092
cites work: The Witt invariant of the form \(\text{Tr}(x^ 2)\)
cites work: Frobenius Galois groups over quadratic fields
cites work: SL(2,5) and Frobenius Galois Groups Over Q
cites work: Q-admissibility of sylow-metacyclic groups as quotient group
cites work: On central extensions of \(A_ n\) as Galois group over \({\mathbb{Q}}\)
cites work: Subfields of division rings. I
description: scientific article; zbMATH DE number 4057629