The algebraic nature of the elementary theory of PRC fields
type: Item
instance of: scholarly article
title: The algebraic nature of the elementary theory of PRC fields
zbMATH Open document ID: 0675.12016
DOI: 10.1007/BF01258664
author: Moshe Jarden
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155293
review text: A field K is PAC (resp. PRC) if every nonempty absolutely irreducible variety V defined over K has a K-rational point (resp. if each such V has a K-rational point provided it has a simple \(\bar K\)-rational point for every real closure \(\bar K\) of K). - After \textit{M. Fried} and \textit{M. Jarden} [``Field arithmetic'', Ergebn. Math. Grenzgebiete, 3. Folge 11 (1986; Zbl 0625.12001)], the elementary theory of algebraic PAC fields determines the elementary theory of all PAC fields (thus a sentence \(\theta\) is true in each field of characteristic 0 if it is true in each PAC field which is algebraic over \({\mathbb{Q}}).\) The scope of this paper is to prove an analogous result for PRC and maximal PRC fields. Also is proved that the absolute Galois group of a maximal PRC field is a free product of groups of order 2 in the category of pro-2 groups and conversely, each such group is isomorphic to the absolute Galois group of a maximal PRC field.
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 12L10
zbMATH DE Number: 4105089
zbMATH Keywords: rational point
zbMATH Keywords: elementary theory of algebraic PAC fields
zbMATH Keywords: maximal PRC fields
zbMATH Keywords: absolute Galois group
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4105089