# 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.

## References

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