On integral bases in relative quadratic extensions

Abstract

Let F \mathcal F be an algebraic number field and E \mathcal E a quadratic extension with E = F ( μ ) \mathcal E=\mathcal F(\sqrt {\mu }) . We describe a minimal set of elements for generating the integral elements o E o_{\mathcal E} of E \mathcal E as an o F 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 E \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.