Splitting quaternion algebras over quadratic number fields

Abstract

We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over $\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free integer. The algorithm is randomized and runs in polynomial time if one is allowed to call oracles for factoring integers.

arXiv comment12 pages, revised version, accepted for publication in Journal of Symbolic Computation