Real free groups and the absolute Galois group of \(R(t)\)
type: Item
instance of: scholarly article
title: Real free groups and the absolute Galois group of \(R(t)\)
zbMATH Open document ID: 0577.12019
DOI: 10.1016/0022-4049(85)90093-3
author: Dan Haran
author: Moshe Jarden
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: A profinite group \(G\) is called ``real projective'' if the set \(\text{Inv}(G)\) of involutions of \(G\) is a closed subset of \(G\), and to every epimorphism \(\alpha: B\to A\) of finite groups and every homomorphism \(\phi: G\to A\) such that for every \(\delta\in \text{Inv}(G)\setminus \text{Ker}(\phi)\) there exists an \(\varepsilon\in \text{Inv}(B)\) for which \(\alpha (\varepsilon)=\phi (\delta)\), there exists a homomorphism \(\gamma: G\to B\) such that \(\alpha \circ \gamma =\phi.\) The authors develop the notion of a ``real free'' profinite group in such a way that the real projective groups are exactly the closed subgroups of real free profinite groups. This parallels the fact that projective profinite groups are the closed subgroups of free profinite groups. A profinite group \(D\) is called ``real free'' if it contains disjoint closed subsets \(X\) and \(Y\) such that \(X\subset \text{Inv}(D)\), \(1\in Y\) and every continuous map \(\phi\) from \(X\cup Y\) into a profinite group \(G\), such that \(\phi (x)^ 2=1\) for every \(x\in X\) and \(\phi (1)=1\), extends to a unique homomorphism of \(D\) into \(G\). It is shown that the absolute Galois group of \(R(t)\) is real free for real closed fields \(R\).
reviewed by: Alexander Prestel
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 20E18
zbMATH DE Number: 3922802
zbMATH Keywords: rational function field
zbMATH Keywords: profinite group
zbMATH Keywords: real projective
zbMATH Keywords: real free
zbMATH Keywords: absolute Galois group
zbMATH Keywords: real closed fields
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90093-3
OpenAlex ID: W2088682330
cites work: The topological space of orderings of a rational function field
cites work: The Boolean Space of Orderings of a Field
cites work: Q5524238
cites work: Free pro-\b C-groups
cites work: A Kurosh Subgroup Theorem for Free Pro--Products of Pro--Groups
cites work: Q3726225
cites work: Die Struktur der absoluten Galoisgruppe über dem Körper \(R(t)\)
cites work: Q3340956
cites work: Q4258075
cites work: Elementare Aussagen zur Arithmetik und Galoistheorie von Funktionenkörpern.
description: scientific article; zbMATH DE number 3922802