Nonrigid constructions in Galois theory
type: Item
instance of: scholarly article
title: Nonrigid constructions in Galois theory
zbMATH Open document ID: 0788.12001
author: Michael D. Fried
author: Pierre Dèbes
published in: Pacific Journal of Mathematics
publication date: 1994-06-05T00:00:00Z
review text: This paper considers diophantine problems with realizing a finite group \(G\) as the Galois group of a regular extension \(L/\mathbb{Q}(x)\) of \(\mathbb{Q}\): \(\overline{\mathbb{Q}}\cap L= \mathbb{Q}\). It illustrates, for example, why you can't dictate where to put the branch points (ramified places) of \(L/\mathbb{Q}(x)\). There is an if and only if condition that \(G\) is the group of a Galois regular extension of \(\mathbb{R}(x)\) with only real branch points. It is that involutions generate \(G\) (Theorem 1.1 (a)). If you can't realize \(G\) with real branch points, you certainly can't realize it with rational branch points. A more sophisticated topic produces a group theory test for finding \(L/\mathbb{R}(x)\) with real branch points, regular but not necessarily Galois, with Galois closure over \(\mathbb{C}\) the (geometric) group \(G\). This direction concludes with a profinite topic. Given \(G\) passing this test, is there a totally nonsplit extension of \(G\) that doesn't pass it? Answers use the Universal Frattini Cover (or minimal projective cover) of \(G\) from [\textit{M. Fried} and \textit{M. Jarden}, Field arithmetic (Ergebnisse 11) (1986; Zbl 0625.12001); Chapter 20]. Of course, the real algebraic numbers are close to the algebraic closure. The field of totally real numbers \(\mathbb{Q}^{tr}\) -- algebraic numbers all of whose conjugates are real -- gives closer comparison with regular realization of groups over \(\mathbb{Q}\). Theorem 5.7 uses [\textit{M. Fried} and \textit{H. Völklein}, Ann. Math., II. Ser. 135, 1-13 (1992; Zbl 0765.12002)] and [\textit{F. Pop}, Fields of totally \(\Sigma\)-adic numbers (Preprint 1991)] to show each finite group is the Galois group of a regular extension of \(\mathbb{Q}^{\text{tr}}(x)\). Theorem 5.7 follows trivially if we know \(G(\overline{\mathbb{Q}}/\mathbb{Q}^{\text{tr}})\) -- generated by the conjugate of complex conjugation -- is freely (in a profinite sense) generated by involutions. \textit{M. Fried}, \t…
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 20B30
zbMATH DE Number: 537375
zbMATH Keywords: coverings of \(\mathbb{P}^ 1\)
zbMATH Keywords: Galois groups
zbMATH Keywords: Hurwitz monodromy group
zbMATH Keywords: finite groups
zbMATH Keywords: Galois group of regular extension of \(\mathbb{Q}\)
zbMATH Keywords: universal Frattini cover
zbMATH Keywords: rational point on varieties
zbMATH Keywords: regular realization problem on dihedral groups
zbMATH Keywords: real branch points
zbMATH Keywords: totally nonsplit extension
zbMATH Keywords: field of totally real numbers
zbMATH Keywords: fields of definition
zbMATH Keywords: inverse Galois problem
zbMATH Keywords: symmetric group
zbMATH Keywords: modular curves
MaRDI profile type: Publication
full work available at URL: pjm.1994.163.81
OpenAlex ID: W2017204263
Wikidata QID: Q125017440
DOI: 10.2140/PJM.1994.163.81
description: scientific article; zbMATH DE number 537375