Tensor products of division algebras and fields

This paper began as an investigation of the question of whether D-1 circle times(F) D-2 is a domain where the D-i 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 F-1 circle times(F) F-2 where the F-i 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. (C) 2013 Published by Elsevier Inc.