On integral bases in relative quadratic extensions

Let $\mathcal F$ be an algebraic number field and $\mathcal E$ a quadratic extension with $\mathcal E=\mathcal F(\sqrt {\mu})$. We describe a minimal set of elements for generating the integral elements $o_{\mathcal E}$ of $\mathcal E$ as an $o_{\mathcal F}$ module. A consequence of this theoretical result is an algorithm for constructing such a set. The construction yields a simple procedure for computing an integral basis of $\mathcal E$ as well. In the last section, we present examples of relative integral bases which were computed with the new algorithm and also give some running times.