Tensor products of division algebras and fields

This paper began as an investigation of the question of whether D1⊗FD2D1⊗FD2 is a domain where the DiDi are division algebras and F   is an algebraically closed field contained in their centers. We present an example where the answer is “no”, and also study the Picard group and Brauer group properties of F1⊗FF2F1⊗FF2 where the FiFi are fields. Finally, as part of our example, we have results about division algebras and Brauer groups over curves. Specifically, we give a splitting criterion for certain Brauer group elements on the product of two curves over F.