Residuacity properties of real quadratic units
The main result of the present paper is that if \(m = 5\) or \(13\), and if \(p\) is a rational prime such that \((-1/p) = 1 = (m/p)\), then \([\varepsilon_m/\mathfrak p] = 1\) if and only if \(p=x^2 + 4my^2\) \((x\in \mathbb Z,\ y\in\mathbb Z)\), where \(\varepsilon_m\) denotes the fundamental unit of the field \(K=\mathbb Q(m^{1/2})\) and \([\cdot / \cdot]\) denotes the Legendre symbol of the ring of integers of \(K\) with \(\mathfrak p\) a prime ideal divisor of \(p\) in \(K\).
necessary and sufficient conditions for representing certain classes of primes by given quadratic forms
scientific article; zbMATH DE number 3427485