On the congruences for the class numbers of the quadratic fields whose discriminants are divisible by 8
type: Item
instance of: scholarly article
title: On the congruences for the class numbers of the quadratic fields whose discriminants are divisible by 8
zbMATH Open document ID: 0587.12002
DOI: 10.1016/0022-314X(86)90006-5
author: Mitsuhiko Hikita
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: Let \(m>0\) be a squarefree odd rational integer, having no prime factor \(\equiv 3\) (mod 4). Let \(\epsilon =T+U \sqrt{2m}>1\) be the fundamental unit of the real quadratic field \({\mathbb{Q}}(\sqrt{2m})\). The author proves the following congruence \[ (1)\quad h(-2m)\equiv (\frac{2}{m}) TU h(2m)\quad (mod 2^{r+2}), \] where r is the number of primes dividing m. Unfortunately the author does not indicate in his paper how (1) is related to congruences already in the literature. We just note that the case \(m=p\) (prime) \(\equiv 5\) (mod 8) of (1) is due to the reviewer [Rocky Mt. J. Math. 11, 19-26 (1981; Zbl 0437.12006)] and that stronger congruences than (1) in the case \(m=p\) (prime) \(\equiv 1\) (mod 4) are due to \textit{P. Kaplan} and the reviewer [Acta Arith. 40, 289-296 (1982; Zbl 0491.12003); ibid. 40, 375-397 (1982; Zbl 0513.12007)].
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 3941639
zbMATH Keywords: class number
zbMATH Keywords: quadratic field
zbMATH Keywords: congruences
reviewed by: Kenneth S. Williams
MaRDI profile type: Publication
cites work: Q5792522
cites work: Q4753369
description: scientific article; zbMATH DE number 3941639