Exceptional units and Euclidean number fields
type: Item
instance of: scholarly article
title: Exceptional units and Euclidean number fields
zbMATH Open document ID: 1124.11050
author: Julien Houriet
published in: Archiv der Mathematik
publication date: 2007-06-07T00:00:00Z
full work available at URL: 13 2006 Article 1019.pdf
review text: Let \(K\) be an algebraic number field and \({\mathcal O}^{*}\) be it's group of units. An element \(\varepsilon \in {\mathcal O}^{*}\) is an exceptional unit if and only if \(1-\varepsilon \in {\mathcal O}^{*}\). H. W. Lenstra has shown that if there exist sufficiently many exceptional units with the property that all of their differences are also units, then the number field \(K\) is Euclidean with respect to the norm. Using this result, the author gives in this paper two methods for computing a subset of \({\mathcal O}^{*}\) such that the difference of two elements of this subset is also in \({\mathcal O}^{*}\). In particular, one method is based on graph theory algorithm for the maximum clique problem. His computations yields \(42\) new Euclidean number fields with degrees 8, 9, 10, 11 and 12.
reviewed by: Abdelmalek Azizi
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 5162041
zbMATH Keywords: Exceptional units
zbMATH Keywords: Euclidean number fields, Lenstra's constant
zbMATH Keywords: maximum clique problem
MaRDI profile type: Publication
OpenAlex ID: W2027835503
DOI: 10.1007/S00013-006-1019-0
description: scientific article; zbMATH DE number 5162041