Congruences modulo 8 for class numbers of general quadratic fields \(\mathbb{Q}(\sqrt{m})\) and \(\mathbb{Q}(\sqrt{-m})\)
type: Item
instance of: scholarly article
title: Congruences modulo 8 for class numbers of general quadratic fields \(\mathbb{Q}(\sqrt{m})\) and \(\mathbb{Q}(\sqrt{-m})\)
zbMATH Open document ID: 0693.12005
DOI: 10.1016/0022-314X(89)90089-9
author: Xianke Zhang
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: The author proves known congruences modulo 8 between class numbers and units of the two fields \(\mathbb{Q}(\sqrt{\pm m})\) [\textit{P.-J. Desnoux}, Manuscr. Math. 62, No. 2, 163--179 (1988; Zbl 0664.12002) and other]. Remarks: (i) Now such congruences are true modulo \(2^{r+2}\) (at least), where \(r\) is the number of odd prime divisors of \(m\). (ii) As many papers are written on the subject, the reviewer takes this opportunity to give less known references: \textit{P.-J. Desnoux} [loc. cit.]; \textit{G. Gras} [Manuscr. Math. 57, 373--415 (1987; Zbl 0599.12016) and Acta Arith. 52, No. 2, 147--162 (1989; Zbl 0618.12004)]; \textit{A. Costa} [Modular forms and class number congruences (Ph. D. Thesis), Univ. Pennsylvania (1989)]; \textit{R. Pioui} [Mesures de Haar \(p\)-adiques et interprétation arithmétique de \(L_2(\chi,s) - L_2(\chi,t)\), \(s,t\in\mathbb{Z}_2\) (Thèse), Univ. Besançon (1990)].
reviewed by: Georges Gras
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 4134168
zbMATH Keywords: class number
zbMATH Keywords: congruences
zbMATH Keywords: quadratic fields
zbMATH Keywords: units
MaRDI profile type: Publication
cites work: Q3943047
cites work: On a class number relation of imaginary Abelian fields
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cites work: On the congruences for the class numbers of the quadratic fields whose discriminants are divisible by 8
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description: scientific article; zbMATH DE number 4134168