Genus fields of real biquadratic fields (Q2270195)

Let \(K\) be a number field and \(Cl(K)\) its class group. Let \(H\) be the Hilbert class field of \(K\), that is the maximal abelian unramified extension of \(K\). Let \(G=\text{Gal}(H/K)\) (\(\simeq \text{Cl}(K)\)). The Hilbert genus field of \(K\) is the subfield of \(H\) fixed by \(G^2\). Since \(\text{Gal}(E/K)\simeq G/G^2\) is an abelian extension of exponent \(2\), by Kummer theory \(E=K(\sqrt{\Delta})\), where \(K^{*2}\subset \Delta\subset K^*\) and \(K^*=K\setminus\{0\}\). The main result of the paper gives an explicit description of \(E\) (by giving an explicit \(\Delta\)) when \(K=\mathbb{Q}(\sqrt{p}, \sqrt{d})\), where \(p\) is a prime number and \(d\) is a squarefree positive integer such that \(p\equiv 1\bmod 4\) and \(d\equiv 3\bmod 4\). The explicit \(\Delta\) uses solutions of certain Diophantine equations coming from Legendre symbols. This result is a generalization of that one in [\textit{P.~J. Sime}, J. Number Theory 50, No. 1, 154--166 (1995; Zbl 0821.11058)].