An elementary proof of Gauss' genus theorem
type: Item
instance of: scholarly article
title: An elementary proof of Gauss' genus theorem
zbMATH Open document ID: 0763.11042
author: Hideo Wada
author: Fidel Nemenzo
published in: Proceedings of the Japan Academy. Series A
publication date: 1993-01-17T00:00:00Z
review text: Let \(m\neq 1\) be a square-free integer and \(d\) the discriminant of the quadratic field \(K=\mathbb{Q}(\sqrt m)\). Let \(p_ 1,\dots,p_ s\) be the odd prime divisors of \(n\). If \(A\) and \(B\) are ideals in \(K\) such that \(A=(\rho)B\) and \(N_ \rho>0\) we write \(A\sim B\). Gauss' duplication theorem asserts that if \(A\) is an ideal coprime with \(d\) then \(A\sim B^ 2\) for some ideal \(B\) if and only if \(\left({NA\over p_ i}\right)=1\), \(1\leq i\leq s\). The authors give a beautiful elementary proof of this famous result, which makes use of Legendre's familiar necessary and sufficient condition for the equation \(ax^ 2+by^ 2=z^ 2\) to have a non-trivial integral solution. The proof only takes about twenty lines and one's reaction is ``Now why didn't I think of that?'' (The ideal \(A\) should be nonzero in Proposition 1).
reviewed by: Kenneth S. Williams
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 97937
zbMATH Keywords: ideal class group
zbMATH Keywords: genera
zbMATH Keywords: quadratic field
zbMATH Keywords: ideals
zbMATH Keywords: Gauss' duplication theorem
zbMATH Keywords: elementary proof
MaRDI profile type: Publication
full work available at URL: pjaa.68.94
OpenAlex ID: W2018057544
cites work: Q3939864
cites work: Comparaison des $2$-groupes des classes d'idéaux au sens large et au sens étroit d'un corps quadratique réel
cites work: Tables of ideal class groups of real quadratic fields
DOI: 10.3792/PJAA.68.94
description: scientific article; zbMATH DE number 97937