Relative integral bases for quartic fields over quadratic subfields (Q1912687)

Let \(K= \mathbb{Q} (\sqrt {c})\) be a quadratic number field and \(L= \mathbb{Q}( \sqrt {a+ b \sqrt {c}})\) be a quadratic extension of \(K\). Here \(a\), \(b\) and \(c\) can be chosen to be rational integers with \(c\) and \((a, b)\) both square-free. Set \((a+ b\sqrt {c})= S^2 R\) where \(R\) and \(S\) are integral ideals of \(K\) with \(R\) square-free. In an earlier article [J. Number Theory 51, 87-102 (1995; Zbl 0826.11048)], \textit{J. G. Huard} and the authors determined the absolute discriminant of \(L\), as well as an integral basis for \(L\). Here the relative discriminant is shown to have the form \(T^2 R\) where \(T\) is an integral divisor of (2) in \(K\). Moreover, it is shown that \(L\) has a relative integral basis over \(K\) if and only if \(S= T(\gamma)\) for some nonzero \(\gamma\) in \(K\). When a relative integral basis exists it is explicitly determined and shown to have one of seven different forms.