\(k\)-reduced ideals of orders of real quadratic fields
type: Item
instance of: scholarly article
title: \(k\)-reduced ideals of orders of real quadratic fields
zbMATH Open document ID: 0847.11056
author: Pierre Kaplan
published in: Journal of the Mathematical Society of Japan
publication date: 1996-10-13T00:00:00Z
review text: Eisenstein sought a priori criteria for determining when \((*)\) \(|x^2- Dy^2|= 4\) has integer solutions where \(\text{gcd}(x, y)= 1\) and \(D\) is a fundamental discriminant. Authors, such as \textit{A. J. Stephens} and \textit{H. C. Williams} [in: Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 869-886 (1989; Zbl 0689.10024)], used the infrastructure of a real quadratic field to find an effective algorithm for determining when \((*)\) has solutions. Note that, when \(D\equiv 5\pmod 8\), the existence of solutions to \((*)\) is equivalent to the fundamental unit of the maximal order \({\mathcal O}_D\) in \(\mathbb{Q}(\sqrt D)\) not being in the non-maximal order \({\mathcal O}_{4D}\), of index 2 in \({\mathcal O}_D\). In the paper under review, the author uses homomorphisms between orders, and defines a grading for reduced ideals. With these tools in hand, he gives criteria for the solution of \((*)\) (too technical to state here). Other results are also presented.
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 844448
zbMATH Keywords: Pell's equation
zbMATH Keywords: real quadratic field
zbMATH Keywords: homomorphisms between orders
zbMATH Keywords: grading for reduced ideals
reviewed by: Richard A. Mollin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04710171
OpenAlex ID: W2087105081
DOI: 10.2969/JMSJ/04710171
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description: scientific article; zbMATH DE number 844448