Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of \(\mathbb{Q}\)
type: Item
instance of: scholarly article
title: Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of \(\mathbb{Q}\)
zbMATH Open document ID: 0816.11066
DOI: 10.1016/0168-0072(94)90024-8
author: Alexandra Shlapentokh
published in: Annals of Pure and Applied Logic
publication date: 1994-08-29T00:00:00Z
review text: Let \(R\) be a subring of the field of algebraic numbers. The analogue of Hilbert's tenth problem for \(R\) is the question whether there is an algorithm which determines if an arbitrary given diophantine equation (with coefficients in \(R\)) has solutions in \(R\). For all known cases where \(R\) is a finitely generated \(\mathbb{Z}\)-module the problem has a negative answer and similar results are known in cases where \(R\) is a ring of ``\(S\)-integers'' of a number field (integers over a product of powers of a fixed finite set of primes). In contrast if \(R\) is the ring of all algebraic numbers the problem has a positive answer. The paper provides examples of rings \(R\) which are rings of \(S\)-integers of infinite extensions of \(\mathbb{Q}\) where Hilbert's tenth problem has a negative answer.
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 11D99
Mathematics Subject Classification ID: 03B25
zbMATH DE Number: 626269
zbMATH Keywords: analogue of Hilbert's tenth problem
zbMATH Keywords: diophantine equation
zbMATH Keywords: ring of all algebraic numbers
zbMATH Keywords: positive answer
reviewed by: Thanases Pheidas
MaRDI profile type: Publication
full work available at URL: 0168-0072(94)90024-8
OpenAlex ID: W2070490082
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description: scientific article; zbMATH DE number 626269