A remark on the ring of algebraic integers in \(\mathbb{Q}(\sqrt {- d})\)
type: Item
instance of: scholarly article
title: A remark on the ring of algebraic integers in \(\mathbb{Q}(\sqrt {- d})\)
zbMATH Open document ID: 1365.11118
author: Wen-Yao Chang
author: Chih-Ren Cheng
author: Ming-Guang Leu
published in: Israel Journal of Mathematics
publication date: 2017-01-11T00:00:00Z
review text: Let \(D\) be an integral domain. If \(a,b\in D\) and there exist elements \(q_1,r_1,q_2,r_2,\dots,r_{n-1},q_n\in D\) with \[ a=q_1b+r_1,b=q_2r+1+r_2,\dots,r_i=q_{i+2}r_{i+1}+r_i,\dots, r_{n-2}=q_nr_{n-1}, \] then one says that the pair \((a,b)\) has a terminating division chain. If this happens for all \(a,b\in D\), then \(D\) is called a quasi-Euclidean domain. It has been shown by \textit{O. T. O'Meara} [J. Reine Angew. Math. 217, 79--108 (1965; Zbl 0128.25502)] that \(D\) is quasi-Euclidean if and only if \(D\) is a \(GE_2\)-ring and a Bezout domain. The authors prove (Theorem 3.2) that every pair \(a,b\in D\) generating \(D\) has such a chain if and only if \(D\) is a \(GE_2\)-ring, and for every non-Euclidean ring of quadratic integers which is a unique factorization domain (i.e., rings of integers of \(\mathbb Q(\sqrt{-d})\) with \(d=19, 43, 67, 163\)) they construct pairs of elements for which there is no finite division chain.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 13F07
zbMATH DE Number: 6673865
zbMATH Keywords: Euclidean domains
zbMATH Keywords: quasi-Euclidean domains
zbMATH Keywords: quadratic fields
zbMATH Keywords: \(GE_2\)-rings
MaRDI profile type: Publication
full work available at URL: s11856-016-1421-7
OpenAlex ID: W2530285660
cites work: Q4138827
cites work: The 2-stage Euclidean algorithm and the restricted Nagata's pairwise algorithm
cites work: On a proposition of Samuel and 2-stage Euclidean algorithm in global fields
cites work: Q4320503
cites work: On the structure of the \(GL_ 2\) of a ring
cites work: A weakening of the euclidean property for integral domains and applications to algebraic number theory. I.
cites work: Quasi-Euclidean subrings of ℚ[x]
cites work: ℤ[] is Euclidean
cites work: Euclidean Rings of Algebraic Integers
cites work: Euklidischer Algorithmus und die Gruppe \(GL_2\)
cites work: The Euclidean algorithm
cites work: On the finite generation of linear groups over Hasse domains.
cites work: Q4773795
DOI: 10.1007/S11856-016-1421-7
description: scientific article; zbMATH DE number 6673865