Euclidean real quadratic number fields
type: Item
instance of: scholarly article
title: Euclidean real quadratic number fields
zbMATH Open document ID: 0564.12003
DOI: 10.1007/BF01235777
author: David H. Johnson
author: Clifford S. Queen
author: A. Sevilla
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let \(K\) be a real quadratic field and let \({\mathcal O}_ K\) denote the ring of integers of \(K\). The authors call \(K\) ``Generalized Euclidean'' (= G.E.) iff given \(\alpha\),\(\beta\in {\mathcal O}_ K\), \(\beta\neq 0\) and the ideal \((\alpha,\beta)\) principal, there exist \(\gamma,\delta\in {\mathcal O}_ K\) such that \(\alpha =\beta \delta +\gamma\), \(| N(\gamma)| <| N(\beta)|\). The following results are proved: (1) If \(d>1\) is square-free and \(d\not\equiv 1 \bmod 4\), then \(K=\mathbb Q(\sqrt{d})\) is G.E. only if \(K\) is Euclidean (with respect to the norm) except when \(d=10\). (2) If \(K=\mathbb Q(\sqrt{p})\), where \(p\) ranges over all primes \(\equiv 1\bmod 4\), then \(K\) is G.E. for only finitely many \(p\).
reviewed by: Veikko Ennola
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 3899011
zbMATH Keywords: Euclidean algorithm
zbMATH Keywords: real quadratic field
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01235777
OpenAlex ID: W2005938594
description: scientific article; zbMATH DE number 3899011