The construction of SL(2,3)-polynomials
type: Item
instance of: scholarly article
title: The construction of SL(2,3)-polynomials
zbMATH Open document ID: 0566.12003
DOI: 10.1016/0022-314X(84)90080-5
author: Franz-Peter Heider
author: Paulgerd Kolvenbach
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: In this paper polynomials over \({\mathbb{Z}}\) with Galois group the special linear group SL(2,3) are constructed. Since SL(2,3) is the nontrivial extension of the alternating group \(A_ 4\) with kernel \({\mathbb{Z}}/2{\mathbb{Z}}\), the realization of SL(2,3) as a Galois group over \({\mathbb{Q}}\) can be considered as an embedding problem. Let f(X)\(\in {\mathbb{Q}}[X]\) be a quartic polynomial with Galois group \(A_ 4\). The authors prove that, under some conditions on f, the Galois group of \(F(X)=f(X^ 2)\) is isomorphic either to SL(2,3) or \(C_ 2\times A_ 4\), or a certain group of order 96. Next, they give a computational criterion to decide whether this Galois group is isomorphic to SL(2,3). Totally real quartic polynomials f(X) over \({\mathbb{Q}}\) with Galois group \(A_ 4\) are constructed and for some of them it is checked that the Galois group of \(F(X)=f(X^ 2)\) is SL(2,3).
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 11R80
Mathematics Subject Classification ID: 12E12
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 3902774
zbMATH Keywords: constructive Galois theory
zbMATH Keywords: SL(2,3) as Galois group
zbMATH Keywords: inverse problem of Galois theory
zbMATH Keywords: polynomials
zbMATH Keywords: embedding problem
zbMATH Keywords: Totally real quartic polynomials
reviewed by: Nuria Vila
MaRDI profile type: Publication
cites work: On the Euler and Bernoulli polynomials.
cites work: Sylow-metacyclic groups and Q-admissibility
cites work: Méthodes et algorithmes pour le calcul numérique du nombre de classes et de unités des extensions cubiques cycliques de Q.
cites work: Endliche Gruppen I
cites work: The genus fields of algebraic number fields
cites work: Q5585021
cites work: Q4135671
cites work: Computation of Polya Polynomials of Primitive Permutation Group
cites work: Q5584592
cites work: Q5569508
cites work: Equidistributed permutation groups
cites work: Berechnung kleiner Diskriminanten total reeller algebraischer Zahlkörper.
cites work: The Simplest Cubic Fields
cites work: Q4129540
cites work: The Determination of Galois Groups
cites work: Arithmetik in Frobeniuserweiterungen
cites work: Q5511421
cites work: Q5659661
description: scientific article; zbMATH DE number 3902774