Efficient computations in central simple algebras using Amitsur cohomology

We introduce a presentation for central simple algebras over a field k using Amitsur cohomology. We provide efficient algorithms for computing a cocycle corresponding to any such algebra given by structure constants. If k is a number field, we use this presentation to prove that the explicit isomorphism problem (i.e., finding an isomorphism between central simple algebras given by structure constants) reduces to S-unit group computation and other related number theoretical computational problems. This also yields, conditionally on the generalised Riemann hypothesis, the first polynomial quantum algorithm for the explicit isomorphism problem over number fields.