On integral bases in relative quadratic extensions

Daberkow, M.; Pohst, M.

doi:10.1090/S0025-5718-96-00686-2

On integral bases in relative quadratic extensions
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by M. Daberkow and M. Pohst PDF

Math. Comp. 65 (1996), 319-329 Request permission

Abstract:

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.

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