Class number 1 criteria for real quadratic fields of Richaud-Degert type
type: Item
instance of: scholarly article
title: Class number 1 criteria for real quadratic fields of Richaud-Degert type
zbMATH Open document ID: 0846.11060
author: Dongho Byeon
author: Hyun Kwang Kim
published in: Journal of Number Theory
publication date: 1996-09-26T00:00:00Z
review text: The real quadratic field \(\mathbb{Q} (\sqrt d)\) is called ``Richaud-Degart type'' if in the unique expression \(d = n^2 + r\) \((- n < r \leq n)\), \(r \mid 4n\) holds. In this paper, the authors intend to provide various class number one criteria for real quadratic fields of Richaud-Degert type by using two ways of computing special values of the zeta function \(\zeta(s)\) attached to the real quadratic field. In one of them, which is obtained by specializing Siegel's formula, the value \(\zeta (-1)\) is expressed in terms of the sum of divisors of integers. In another way, which is due to \textit{H. Lang} [cf. J. Reine Angew. Math. 233, 123-175 (1968; Zbl 0165.36504)], the value \(\zeta (-1, {\mathcal A})\) corresponding to an ideal class \({\mathcal A}\) is expressed in terms of the generalized Dedekind sums. By comparing these values they obtain various class number one criteria, which are expressed in terms of \(n\) and \(r\), for real quadratic fields of Richaud-Degert type.
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 874200
zbMATH Keywords: special values of the zeta function
zbMATH Keywords: Siegel's formula
zbMATH Keywords: class number one
zbMATH Keywords: real quadratic fields of Richaud-Degert type
reviewed by: Hideo Yokoi
MaRDI profile type: Publication
full work available at URL: jnth.1996.0052
OpenAlex ID: W2983437486
DOI: 10.1006/JNTH.1996.0052
description: scientific article; zbMATH DE number 874200