Splitting quaternion algebras over quadratic number fields

TL;DR

This paper introduces a randomized polynomial-time algorithm for finding zero divisors in quaternion algebras over quadratic number fields, effectively solving certain quadratic equations over these fields, assuming access to integer factorization oracles.

Contribution

It presents the first efficient randomized algorithm for splitting quaternion algebras over quadratic fields, linking algebraic problems to quadratic equation solving.

Findings

Algorithm runs in polynomial time with factoring oracles

Successfully finds zero divisors in quaternion algebras over quadratic fields

Reduces algebraic problem to solving quadratic equations over 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.