A refinement of the Dress-Scharlau theorem
type: Item
instance of: scholarly article
title: A refinement of the Dress-Scharlau theorem
zbMATH Open document ID: 1331.11092
author: Se Wook Jang
author: Byeong Moon Kim
published in: Journal of Number Theory
publication date: 2015-09-09T00:00:00Z
review text: Let \(K_d=\mathbb{\mathbb Q}(\sqrt{d})\) be a real quadratic field. For \(\alpha=a+b\sqrt{d} \in K_d\), let \(\overline{\alpha}=a-b\sqrt{d}\) be the conjugate of \(\alpha\). Let \(\mathcal{O}_{d}^{\pm}\) be the set of \(\alpha \in \mathcal{O}_d\) satisfying \(\alpha>0\) and \(\overline{\alpha}<0\). The authors show that if \(\alpha \in \mathcal{O}_{d}^{\pm}\) and the norm \(|\alpha \overline{\alpha}|\) is greater than \(d\) if \(d \equiv 2,3 \pmod 4\) (resp. greater than \((d-1)/4\) if \(d \equiv 1 \pmod 4\)) then \(\alpha\) is decomposable in \(\mathcal{O}_{d}^{\pm}\), i.e., there exist \(\beta\) and \(\gamma \in \mathcal{O}_{d}^{\pm}\) such that \(\alpha=\beta+\gamma\) (Theorem 4). Other results of this type are also obtained.
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 6479848
zbMATH Keywords: continued fractions
zbMATH Keywords: totally positive integers
zbMATH Keywords: real quadratic fields
reviewed by: A. Dubickas
MaRDI profile type: Publication
full work available at URL: j.jnt.2015.06.003
OpenAlex ID: W1123341274
cites work: Indecomposable totally positive numbers in real quadratic orders
DOI: 10.1016/J.JNT.2015.06.003
description: scientific article; zbMATH DE number 6479848