On a conjecture of Brandler
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 \(\mathbb Q((-m)^{1/2})\).
quadratic residuacity properties of real quadratic units
scientific article; zbMATH DE number 3557892