A Non-commutativity Statement for Algebraic Quaternions

In this paper we provide a constructive version of Tits alternative for a broad class of quaternions with algebraic coefficients. Our result is a generalization of that contained in the paper [1], concerning groups of rational quaternions. Indeed, the tools developed in [1] can be extended to arbitrary number fields by translating them in the corresponding Dedekind domain, as the techniques involved are of a typical “factorization and divisibility” flavour. Let K be a finite extension of Q. We will say that a quaternion a+ bi+ cj + dk is K-rational if its coefficients a, b, c, d all lie in K. The main result in the paper is then the following.