The elementary theory of \(e\)-free PAC domains
type: Item
instance of: scholarly article
title: The elementary theory of \(e\)-free PAC domains
zbMATH Open document ID: 0980.03009
DOI: 10.1016/S0168-0072(99)00023-8
author: Aharon Razon
published in: Annals of Pure and Applied Logic
publication date: 2002-02-27T00:00:00Z
review text: In the author's words ``We prove that the theory of all sentences in the language of rings which are true in \({\widetilde {\mathbf Z}}\cap {\widetilde {\mathbf Q}}(\sigma)\) for almost all \(\sigma\) is decidable.'' Here \({\widetilde {\mathbf Q}}\) is the field of algebraic numbers, \({\widetilde {\mathbf Z}}\) is the ring of algebraic integers, \(\sigma\) is an element of the absolute Galois group of \({\mathbf Q}\) and \({\widetilde {\mathbf Q}}(\sigma)\) is the subfield of \({\widetilde {\mathbf Q}}\) which is fixed by \( \sigma\); the phrase ``almost all'' is meant in the sense of the Haar measure. This result should be compared with the results of Rumely, Van den Dries and Macintyre, according to which, the theory of \({\widetilde {\mathbf Z}}\) is decidable. In the opposite direction, due to work of J. Robinson, the theory of the ring of totally real algebraic integers is undecidable. The methods have their origin in work on Pseudo Algebraically Closed (PAC) fields, presented in the book ``Field arithmetic'' [Berlin etc.: Springer (1986; Zbl 0625.12001)] by \textit{M. D. Fried} and \textit{M. Jarden}.
reviewed by: Thanases Pheidas
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 12L05
Mathematics Subject Classification ID: 12L10
zbMATH DE Number: 1448518
zbMATH Keywords: decidability
zbMATH Keywords: algebraic numbers
zbMATH Keywords: algebraic integers
zbMATH Keywords: Galois group
zbMATH Keywords: Haar measure
zbMATH Keywords: pseudo algebraically closed domains
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1448518