On a conjecture of Brandler
type: Item
instance of: scholarly article
title: On a conjecture of Brandler
zbMATH Open document ID: 0358.12003
DOI: 10.1016/0022-314X(76)90094-9
author: Charles J. Parry
published in: Journal of Number Theory
publication date: 1976-01-01T00:00:00Z
review text: In a recent article [J. Number Theory 5, 271--286 (1973; Zbl 0272.12002)], \textit{J. A. Brandler} made a conjecture concerning the quadratic residuacity properties of real quadratic units. In this article Brandler's conjecture is shown to be false and a revised conjecture is stated and proved. The correct version is: Theorem: Let \(m\equiv 1\pmod 8\) be a positive rational prime and let \(\varepsilon_m\) denote the fundamental unit of \(\mathbb Q(m^{1/2})\). Furthermore let \(p\) be an odd prime satisfying \((- 1/p) = 1 = (m/p)\) and let \(\mathcal P\) and \(\hat \mathcal P\) denote prime ideal divisors of \(p\) in \(\mathbb Q(m^{1/2})\) and \(\mathbb Q((-m)^{1/2})\), respectively. If \(h\) denotes the 2-class number of \(\mathbb Q((- m)^{1/2})\) and \(\ell=h/4\) then the quadratic character \([\varepsilon_m/p]=1\) if and only if \(\hat\mathcal P^\ell\) belongs to an ideal class of odd order in \(\mathbb Q((-m)^{1/2})\).
reviewed by: Charles J. Parry
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 3557892
zbMATH Keywords: quadratic residuacity properties of real quadratic units
zbMATH Keywords: revised Brandler's conjecture
MaRDI profile type: Publication
Wikidata QID: Q122945537
cites work: Die 2-Klassenzahlen spezieller quadratischer Zahlkörper.
cites work: Residuacity properties of real quadratic units
cites work: Q4063245
cites work: Q5612629
description: scientific article; zbMATH DE number 3557892