Realization of the Mathieu groups \(M_{11}\) and \(M_{12}\) as Galois groups over \({\mathbb Q})\).
type: Item
instance of: scholarly article
title: Realization of the Mathieu groups \(M_{11}\) and \(M_{12}\) as Galois groups over \({\mathbb Q})\).
zbMATH Open document ID: 0598.12007
DOI: 10.1016/0022-314X(86)90089-2
author: Andreas Zeh-Marschke
author: B. Heinrich Matzat
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: The authors prove that the Mathieu groups \(M_{11}\) and \(M_{12}\) occur as Galois groups over \({\mathbb Q}(t)\). Moreover they construct a polynomial over \({\mathbb Q}(t)\) with Galois group \(M_{12}\) and compute specializations \(t\mapsto \tau \in {\mathbb Q}\) which realize \(M_{12}\) as Galois group over \({\mathbb Q}\). \(M_{12}\) is known to occur as Galois group over \({\mathbb Q}(t)\) and \({\mathbb Q}\) by work of the first author [J. Algebra 96, 499--531 (1985; Zbl 0587.12004)]. Using the same methods, but another class structure for \(M_{12}\) the present results on \(M_{12}\) and \(M_{11}\) are obtained; here the fixed field of the subgroup \(M_{11}\) of \(M_{12}=\text{Gal}(N/{\mathbb Q}(t))\) happens to be a rational function field \({\mathbb Q}(x)\). Constructing a polynomial for \(M_{12}\) leads to a nonlinear system of 13 equations in 12 unknowns over a quadratic extension field of \({\mathbb Q}\) which is solved on a computer with Buchberger's algorithm. A polynomial and specializations for \(M_{11}\) are announced by the authors.
reviewed by: Ulrich Schoenwaelder
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 3963969
zbMATH Keywords: divisor
zbMATH Keywords: inverse problem of Galois theory
zbMATH Keywords: algebraic function field
zbMATH Keywords: Mathieu groups
zbMATH Keywords: \(M_{11}\)
zbMATH Keywords: \(M_{12}\)
zbMATH Keywords: \(M_{12}\) as Galois group over \({\mathbb Q}\)
zbMATH Keywords: rational function field
zbMATH Keywords: Buchberger's algorithm
zbMATH Keywords: specializations
MaRDI profile type: Publication
cites work: Konstruktion von Zahlkörpern mit der Galoisgruppe \(M_{11}\) über \(\mathbb{Q}(\sqrt{-11})\)
cites work: Konstruktion von Zahlkörpern mit der Galoisgruppe \(M_{12}\) über \(Q(\sqrt{-5})\)
cites work: Konstruktion von Zahl- und Funktionenkörpern mit vorgegebener Galoisgruppe.
cites work: Zwei Aspekte konstruktiver Galoistheorie. (Two aspects of constructive Galois theory)
cites work: Zum Einbettungsproblem der algebraischen Zahlentheorie mit nicht abelschem Kern
cites work: On improving approximate results of Buchberger's algorithm by Newton's method
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description: scientific article; zbMATH DE number 3963969