On the normalizer of finitely generated subgroups of absolute Galois groups of uncountable Hilbertian fields of characteristic 0
type: Item
instance of: scholarly article
title: On the normalizer of finitely generated subgroups of absolute Galois groups of uncountable Hilbertian fields of characteristic 0
zbMATH Open document ID: 0667.12008
DOI: 10.1007/BF02778037
author: Wulf-Dieter Geyer
author: Moshe Jarden
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Main topic is the structure of absolute Galois groups of Hilbertian fields, i.e. fields where Hilbert's irreducibility theorem holds true. It is shown that if such fields contain only finitely many roots of unity, then for almost all (in the sense of Haar measure) automorphisms \(\delta_ 1,...,\delta_ e (e\geq 2)\) of the absolute Galois group G of K, the centralizer \(C_ G(<\delta >)\) in G of the subgroup generated by \(\delta_ 1,...,\delta_ e\) is trivial. On the contrary, if K contains all roots of unity, this is no longer true in general. For the Hilbertian rational field \(K=K_ 0(T)\) of characteristic zero with \(\aleph_ 1\) algebraically independent elements over \(K_ 0\), the subset of elements \(\delta_ 1,...,\delta_ e\) of \(G\times G\times...\times G\) such that \(C_ G(<\delta >)=1\) (or \(N_ G(<\delta >)=<\delta >)\) is not even measurable.
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12E99
zbMATH DE Number: 4091622
zbMATH Keywords: absolute Galois groups of Hilbertian fields
MaRDI profile type: Publication
cites work: Q5622313
cites work: Some properties of the absolute Galois group of a Hilbertian field
cites work: Q3027108
cites work: Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist
cites work: Algebraic extensions of finite corank of Hilbertian fields
cites work: Q4063245
cites work: Q5612629
cites work: Q5588717
cites work: Q5822841
cites work: Q5625306
description: scientific article; zbMATH DE number 4091622