On relative integral bases for pure quartic fields
Let \(K=\mathbb{Q}[z]\), \(z^ 4\in\mathbb{Q}\), be a pure quartic field with Galois closure \(L\). For any quadratic extension \(L/L_ 1\), the authors determine whether \(L/L_ 1\) has an integral basis, and similarly for any quadratic extension \(K/K_ 1\). The results are based on a criterion of \textit{H. B. Mann} [Proc. Am. Math. Soc. 9, 167-172 (1958; Zbl 0081.266)]: \(E/F\), quadratic, has an integral basis iff the discriminant ideal \(\Delta_{E/F}\) is principal, \(\Delta_{E/F}=(D)\) for some \(D\) in \(F\) such that \(E=F(\sqrt D)\).
scientific article; zbMATH DE number 68887