Definability of the ring of integers in pro-\(p\) Galois extensions of number fields
type: Item
instance of: scholarly article
title: Definability of the ring of integers in pro-\(p\) Galois extensions of number fields
zbMATH Open document ID: 0984.03034
DOI: 10.1007/BF02803513
author: Carlos Videla
published in: Israel Journal of Mathematics
publication date: 2001-01-25T00:00:00Z
review text: Let \(K\) be a number field and \(M\) a field of algebraic numbers (in a fixed algebraic closure of \(\mathbb Q\)) which extends \(K\). Let \(\widetilde {\mathbb Z}\) be the ring of all algebraic integers. The author proves that if \(M\) is pro-\(p\) for some prime number \(p\), then the ring of algebraic integers of \(M\), \(M\cap \widetilde {\mathbb Z}\), is first-order definable in \(M\), with parameters in \(M\). The methods of proof have their origin in previous relevant work of the author and builds on classical definability results of Rumely (who extended similar results of J. Robinson). The motivation of the paper is to investigate the decidability question for the fields \(M\). This relates to decidability of fields of algebraic numbers which relate to geometrical constructions (e.g. fields of complex or totally real numbers constructible by ruler and compass). The bibliography contains references on several such classical results. Due to results of Rumely, van den Dries and Macintyre on the decidability of the theory of \(\widetilde {\mathbb Z}\), the results of the paper do not suffice to show undecidability of the fields \(M\). It seems strange that, even today, many of the fundamental and intriguing questions which motivate this line of research, are still open.
reviewed by: Thanases Pheidas
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 11U09
zbMATH DE Number: 1506579
zbMATH Keywords: definability
zbMATH Keywords: numbers constructible by rulers and compass
zbMATH Keywords: ring of algebraic integers
zbMATH Keywords: decidability
MaRDI profile type: Publication
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cites work: Metamathematical problems
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full work available at URL: https://doi.org/10.1007/bf02803513
OpenAlex ID: W1132965784
description: scientific article; zbMATH DE number 1506579