Polynomials for primitive nonsolvable permutation groups of degree d\(\leq 15\)
type: Item
instance of: scholarly article
title: Polynomials for primitive nonsolvable permutation groups of degree d\(\leq 15\)
zbMATH Open document ID: 0632.12012
DOI: 10.1016/S0747-7171(87)80056-1
author: Gunter Malle
published in: Journal of Symbolic Computation
publication date: 1987-01-01T00:00:00Z
review text: All nonsolvable groups having a faithful primitive permutation representation of degree \(\leq 15\) are known to occur as Galois groups over \({\mathbb{Q}}\). A constructive method of computing polynomials having one of these groups as Galois groups has been given by \textit{B. H. Matzat} [J. Reine Angew. Math. 349, 179-220 (1984; Zbl 0555.12005)]. Previous results obtained by Matzat (loc. cit.), the author and \textit{B. H. Matzat} [Math. Ann. 272, 549-565 (1985; Zbl 0559.12005)], and \textit{B. H. Matzat} and \textit{A. Zeh-Marschke} [J. Number Theory 23, 195-202 (1986; Zbl 0598.12007)] show that only polynomials with Galois groups \(PGL(2,11)\), \(PSL(3,3)\), \(Hol(E_ 8)\) (the holomorph of an elementary abelian group of order 8) remain to be determined. The author fills in the gap by computing the required polynomials.
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 20F29
zbMATH DE Number: 4027612
zbMATH Keywords: constructive Galois theory
zbMATH Keywords: inverse problem of Galois theory
zbMATH Keywords: Galois groups
zbMATH Keywords: PGL(2,11)
zbMATH Keywords: PSL(3,3)
zbMATH Keywords: \(Hol(E_ 8)\)
reviewed by: Veikko Ennola
MaRDI profile type: Publication
cites work: Realisierung von Gruppen \(PSL_ 2({\mathbb{F}}_ p)\) als Galoisgruppen über \({\mathbb{Q}}\)
cites work: Konstruktion von Zahl- und Funktionenkörpern mit vorgegebener Galoisgruppe.
cites work: Zwei Aspekte konstruktiver Galoistheorie. (Two aspects of constructive Galois theory)
cites work: Realization of the Mathieu groups \(M_{11}\) and \(M_{12}\) as Galois groups over \({\mathbb Q})\).
cites work: On the construction of Galois extensions of function fields and number fields
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description: scientific article; zbMATH DE number 4027612