A relative integral basis over \(\mathbb{Q}(\sqrt{-3})\) for the normal closure of a pure cubic field (Q1811934)

The normal closure of a pure cubic field always has a relative integral basis (RIB) over its quadratic subfield, since the ring of integers of the third cyclotomic field is a PID. In [Int. J. Math. Sci. 9, 97--104 (1986; Zbl 0593.12005)] \textit{M. Haghighi} gave such a RIB under certain conditions. However, in some cases his basis involved finding an element of the pure cubic field of norm 3. This can be difficult to do. In this article the authors give a RIB for \(\mathbb Q(\root 3\of{ab^2},\sqrt{-3})\) over \(\mathbb Q(\sqrt{-3})\) that is a simple rational function of \(\sqrt{-3}\), \(\root 3\of{a^2b}\) for any square free integers \(a\) and \(b\). There are four different cases depending on divisibility conditions of \(a\) and \(b\) by \(3\) and \(a^2- b^2\) by \(9\).