We consider two problems. First let u be an element of a quaternion algebra B over $$\mathbb {Q}\sqrt{d}$$ such that u is non-central and $$u^2\in \mathbb {Q}$$. We relate the complexity of finding an element $$v'$$ such that $$uv'=-v'u$$ and $$v'^2\in \mathbb {Q}$$ to a fundamental problem studied earlier. For the second problem assume that $$A\cong M_2\mathbb {Q}\sqrt{d}$$. We propose a polynomial randomized algorithm which finds a non-central element $$l\in A$$ such that $$l^2\in \mathbb {Q}$$. Our results rely on the connection between solving quadratic forms over $$\mathbb {Q}$$ and splitting quaternion algebras over $$\mathbb {Q}$$ [4], and Castel's algorithm [1] which finds a rational solution to a non-degenerate quadratic form over $$\mathbb {Q}$$ in 6 dimensions in randomized polynomial time. We use these two results to construct a four dimensional subalgebra over $$\mathbb {Q}$$ of A which is a quaternion algebra. We also apply our results to analyze the complexity of constructing involutions.
['Péter Kutas']
Some Results Concerning the Explicit Isomorphism Problem over Number Fields