\(\tilde A_5\) and \(\tilde A_7\) are Galois groups over number fields
type: Item
instance of: scholarly article
title: \(\tilde A_5\) and \(\tilde A_7\) are Galois groups over number fields
zbMATH Open document ID: 0609.12005
DOI: 10.1016/0021-8693(86)90213-9
author: Walter Feit
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: This paper deals with the problem of the realizability of \(\tilde A_n\), the double cover of the alternating group \(A_n\), as Galois group over number fields. This question can be regarded as an embedding problem whose obstruction is related to the Witt invariant of an associated quadratic trace form, by means of \textit{J.-P. Serre}'s trace formula [Comment. Math. Helv. 59, 651--676 (1984; Zbl 0565.12014)]. The author considers generalized Laguerre polynomials \(F_n(X,\lambda,\mu)\in \mathbb Q(\lambda,\mu)[X]\) (cf. [\textit{I. Schur}, Collected Works, Vol. III (1973; Zbl 0274.01054)]) and computes the principal minors of the associated trace form matrix. Then, he shows that for \(n\equiv 3\pmod 4\) the obstruction to the embedding problem in \(\tilde A_ n\) associated to the realization of \(A_n\) given by \(F_n(X,1,1)\in\mathbb Q[X]\) is trivial and, therefore, that for \(n\equiv 3\pmod 4\) \(\tilde A_n\) appears as Galois group over \(\mathbb Q\). The study of suitable Diophantine equations allows the author to prove that there exist infinitely many realizations of \(A_5\) and of \(A_7\) over \(\mathbb Q\) given by Laguerre polynomials which can be embedded in \(\tilde A_5\) and \(\tilde A_7\), respectively. Therefore, \(\tilde A_5\) and \(\tilde A_7\) appear as Galois group over every number field. Let \(K\) be a number field, a finite group \(G\) is called \(K\)-admissible if there exists a Galois extension of \(K\) with Galois group \(G\) which is a maximal subfield of a finite-dimensional division algebra with center \(K\). The author shows that if \(\sqrt{10}\not\in \widehat K(\sqrt{15})\), \(A_5\) is \(K\)-admissible, and if \(\sqrt{10}\not\in \widehat K(\sqrt{3},\sqrt{5},\sqrt{- 2})\), \(\tilde A_5\) is \(K\)-admissible, where \(\widehat K\) denotes the Galois closure of \(K\) over \(\mathbb Q\).
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 3985352
zbMATH Keywords: inverse problem of Galois theory
zbMATH Keywords: Galois groups
zbMATH Keywords: embedding problem
zbMATH Keywords: Witt invariant
zbMATH Keywords: quadratic trace form
zbMATH Keywords: trace formula
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: finite-dimensional division algebra
reviewed by: Nuria Vila
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90213-9
OpenAlex ID: W2000377259
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description: scientific article; zbMATH DE number 3985352