Toward a theorem of Douady
type: Item
instance of: scholarly article
title: Toward a theorem of Douady
zbMATH Open document ID: 1230.12002
published in: Algebra and Logic
publication date: 2011-06-08T00:00:00Z
review text: Let \(F\) be an algebraically closed field of zero characteristic, and let \(\Phi = F(X)\) be a rational function field in one variable over \(F\). As shown by \textit{A. Douady} (see [C. R. Acad. Sci., Paris 258, 5305--5308 (1964; Zbl 0146.42105)]), then the absolute Galois group \(G _ {\Phi }\) is a free profinite group. For that purpose, he has first proved this theorem in the special case where \(F = \mathbb C\), the field of complex numbers (relying on Riemann's existence theorem), and thereafter he has extended it to the general case, making use of Grothendieck's transition method. Alternative transitional techniques relying on model theoretic methods have been used by van den Dries and Ribenboim, for proving Douady's theorem as well as for reproving the Krull-Neukirch theorem concerning \(\mathcal{G}_ {\mathbb R(X)}\) (see \textit{W. Krull} and \textit{J. Neukirch} [Math. Ann. 193, 197--209 (1971; Zbl 0236.12104) and, e.g., \textit{L. van den Dries} and \textit{P. Ribenboim}, Ann. Pure Appl. Logic 33, 83--107 (1987; Zbl 0645.12009)]). The paper under review presents a new model-theoretic transition method, which may also be helpful in describing \(\mathcal{G}_{R(X)}\), where \(R\) is a real closed field. The models considered by the author satisfy a system of elementary conditions of four different types.
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 5905263
zbMATH Keywords: absolute Galois group
zbMATH Keywords: free profinite group
zbMATH Keywords: algebraically closed field
zbMATH Keywords: transition method
zbMATH Keywords: models
zbMATH Keywords: elementary conditions
author: Yuri L. Ershov
MaRDI profile type: Publication
full work available at URL: s10469-011-9113-1
OpenAlex ID: W2043845679
cites work: Q5524238
cites work: Q3861504
cites work: An application of Tarski's principle to absolute Galois groups of function fields
cites work: Die Struktur der absoluten Galoisgruppe über dem Körper \(R(t)\)
cites work: Q5601471
cites work: Q4490702
DOI: 10.1007/S10469-011-9113-1
description: scientific article; zbMATH DE number 5905263