A note on Hasse's theorem concerning the class number formula of real quadratic fields
type: Item
instance of: scholarly article
title: A note on Hasse's theorem concerning the class number formula of real quadratic fields
zbMATH Open document ID: 0537.12003
author: Noriaki Kimura
published in: Tokyo Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let p be a prime with \(p\equiv 1 (mod 4)\) and h be the class number of the real quadratic field \({\mathbb{Q}}(\sqrt{p})\). Let \(\epsilon>1\) be a fundamental unit of \({\mathbb{Q}}(\sqrt{p})\). As is well-known, the Dirichlet class number formula is given in the form \[ \epsilon^ h=\prod_{b}\sin \pi b/p\quad /\prod_{a}\sin \pi a/p\quad, \] where a and b runs over the quadratic residues and quadratic non-residues between 0 and p/2, respectively. \(\epsilon^ h\) is written in the form \(u+v\sqrt{p},\quad u,v\in {\mathbb{Q}}.\) The explicit formula of u and v was given by H. Hasse. In this paper the author proves an alternative form of Hasse's theorem, which is slightly simpler in structure than Hasse's one.
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 3853204
zbMATH Keywords: alternative form of Hasse theorem
zbMATH Keywords: class number of the real quadratic field
zbMATH Keywords: fundamental unit
zbMATH Keywords: Dirichlet class number formula
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270213883
OpenAlex ID: W2035436455
DOI: 10.3836/TJM/1270213883
description: scientific article; zbMATH DE number 3853204