Splitting quaternion algebras over quadratic number fields

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 deterministic and runs in polynomial time if one is allowed to call oracles for factoring integers and polynomials over finite fields.

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