Some finite groups which appear as Gal L/K, where \(K\subseteq {\mathbb{Q}}(\mu _ n)\)
type: Item
instance of: scholarly article
title: Some finite groups which appear as Gal L/K, where \(K\subseteq {\mathbb{Q}}(\mu _ n)\)
zbMATH Open document ID: 0552.12004
DOI: 10.1016/0021-8693(84)90228-X
author: John G. Thompson
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: The topic of this paper is the famous 'inverse problem' of Galois theory: Which groups occur as Galois groups over a fixed ground field k? \(k={\mathbb{Q}}\) or k a small extension of \({\mathbb{Q}}\) are of special interest. The most striking consequence of this work is the realization of the Fischer-Griess monster as a Galois group over \({\mathbb{Q}}.\) The author deals more general with non-abelian finite simple groups F that are 'rigid'. This means: there are conjugacy classes \(C_ 1,...,C_ k\) in F, such that i) F is generated by \(x_ 1,...,x_ k\) for every \((x_ 1,...,x_ k)\in C_ 1\times...\times C_ k\) with \(x_ 1\cdot...\cdot x_ k=1,\) and ii) F operates (by conjugation) transitively on these generators \((x_ 1,...,x_ k).\) This is the fundamental group theoretical concept. Examples of rigid simple groups given by the author are the Fischer-Griess monster and the groups \(PSL(2,2^ n)\). The author then constructs Galois extensions \(L| K\) of some cyclotomic field K with a given rigid group F as Galois group, and more general: Galois extensions \(N| {\mathbb{Q}}\) with a Galois group built out of F in an explicit form. The construction of these field extensions is based on the theory of Riemann surfaces and of discrete subgroups of \(PSL_ 2({\mathbb{R}})\). The proof is rather long and involved, containing a study of the congruence subgroup \(\Gamma_ 0(12)\) with its corresponding modular curve \(X_ 0(12)\), and explicit calculations in terms of Puiseux-series. To fix the ideas behind all this one should add: A group G generated by \(x_ 1,...,x_ k\) \(with\quad x_ 1\cdot...\cdot x_ k=1\) can be realized as the group of covering transformations of some compact Riemann surface \(X\to {\mathbb{P}}^ 1({\mathbb{C}})\), unramified outside k points in \({\mathbb{P}}^ 1({\mathbb{C}})\). Hence G is the Galois group of an algebraic function field L over the field of rational functions \({\mathbb{C}}(T)\). Here \({\mathbb{C}}\) may be replaced by \({\bar {\mathbb{Q}}}…
reviewed by: Norbert Klingen
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 3879038
zbMATH Keywords: inverse problem of Galois theory
zbMATH Keywords: Fischer-Griess monster as Galois group over \({\mathbb{Q}}\)
zbMATH Keywords: finite simple groups
zbMATH Keywords: fundamental group
zbMATH Keywords: rigid simple groups
zbMATH Keywords: cyclotomic field
zbMATH Keywords: discrete subgroups of \(PSL_ 2({\mathbb{R}})\)
zbMATH Keywords: congruence subgroup
zbMATH Keywords: modular curve
zbMATH Keywords: Puiseux-series
zbMATH Keywords: group of covering transformations
zbMATH Keywords: compact Riemann surface
zbMATH Keywords: algebraic function field
zbMATH Keywords: ramification points
zbMATH Keywords: cusps
Wikidata QID: Q55923808
MaRDI profile type: Publication
cites work: Q5835648
cites work: Q5625265
description: scientific article; zbMATH DE number 3879038