Tensor Products of Division Algebras and Fields

This paper began as an investigation of the question of whether $D_1 \otimes_F D_2$isadomainwherethe$D_i$aredivisionalgebrasand$F$ 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 $F_1{\otimes }_F{F}_2$ 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 $F$.