Algorithmic applications of the corestriction of central simple algebras

Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple $L$-algebra. In order to obtain efficient algorithms in the characteristic 2 case, we propose an algorithm for finding nontrivial zeros of a regular quadratic form in four variables over $\mathbb{F}_{2^k}(t)$.