The field of rational numbers is algebraically closed in some of its topological completions
type: Item
instance of: scholarly article
title: The field of rational numbers is algebraically closed in some of its topological completions
zbMATH Open document ID: 0933.12007
DOI: 10.1016/S0022-4049(97)00072-8
published in: Journal of Pure and Applied Algebra
publication date: 2000-03-27T00:00:00Z
review text: The author constructs a class of examples of locally unbounded ring topologies on \(\mathbb{Z}\) (resp. \(k[X]\), \(k\) a field). Similar constructions of field topologies on \(\mathbb{Q}\) and \(k(X)\) are presented. In the case of rings, the completion is an integral domain and it is a field in case of field topologies. It is not clear whether the topologies are minimal or not. Remark: The author's construction can be directly extended to finite field extensions \(K\) of \(\mathbb{Q}\), for which the integral closure of \(\mathbb{Z}\) in \(K\) is UFD.
Mathematics Subject Classification ID: 12J99
zbMATH DE Number: 1308196
zbMATH Keywords: locally unbounded ring topologies
zbMATH Keywords: field topologies
author: José E. Marcos
MaRDI profile type: Publication
cites work: Q3910658
cites work: Integer Topologies
cites work: Q3708229
cites work: Inductive Ring Topologies
cites work: Q5525460
cites work: Q5555075
cites work: On strictly minimal topological division rings
cites work: Topologies on the rational field
cites work: Locally bounded topologies on the rational field
cites work: Q3999309
cites work: Q4104164
cites work: Q3992929
cites work: Locally bounded topologies on Euclidean rings
cites work: Q3994429
cites work: Complete Fields From Local Rings
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description: scientific article; zbMATH DE number 1308196