Some Results Concerning the Explicit Isomorphism Problem over Number Fields

Abstract

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.