A computer algorithm for finding new Euclidean number fields
type: Item
instance of: scholarly article
title: A computer algorithm for finding new Euclidean number fields
zbMATH Open document ID: 0913.11056
author: Roland Quême
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1999-03-15T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93711
full work available at URL: jtnb10-1.html
review text: The main result of this article is the construction of more than 1200 number fields of degrees \(4\), \(5\) and \(6\) that are Euclidean with respect to the absolute value of the norm. The main idea is classical: divide the fundamental domain of the lattice \({\mathcal O}_K\) into sufficiently small cubes and, for each cube, find an element \(\gamma\in O_K\) such that \(| N(\xi-\gamma)|<1\) for all \(\xi\) in that cube. If there are cubes left after several repeated divisions, stage 2 of the algorithm checks whether multiplication by units transforms a cube into some covered region: these cubes can then be eliminated. The author goes on to discuss applications of his algorithm to rings of \(S\)-integers and finally asks whether the inhomogeneous minima of number fields with fixed signature can be bounded by a constant times the square root of the absolute value of the discriminant; as examples in degree \(3\) and \(4\) due to Swinnerton-Dyer and Davenport show (see the reviewer's survey [\textit{F. Lemmermeyer}, Expo. Math. 13, No. 5, 385-416 (1995; Zbl 0843.11046)]), the answer is negative.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11Y16
Mathematics Subject Classification ID: 11R21
zbMATH DE Number: 1229646
zbMATH Keywords: Euclidean algorithm
zbMATH Keywords: Euclidean number fields
zbMATH Keywords: \(S\)-integers
zbMATH Keywords: inhomogeneous minimum
MaRDI profile type: Publication
OpenAlex ID: W2317022136
cites work: The Euclidean Algorithm in Cubic Number Fields
cites work: LINEAR FORMS ASSOCIATED WITH AN ALGEBRAIC NUMBER-FIELD
cites work: The Euclidean algorithm in algebraic number fields
cites work: Euclidean number fields of large degree
cites work: On the finite generation of linear groups over Hasse domains.
cites work: Q5523070
DOI: 10.5802/JTNB.217
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description: scientific article; zbMATH DE number 1229646