Tensor Products of Division Algebras and Fields

TL;DR

This paper explores the properties of tensor products of division algebras and fields, providing counterexamples, and investigates related structures like Picard and Brauer groups, with applications to algebraic curves.

Contribution

It presents a counterexample showing tensor products of division algebras may not be domains and studies Brauer group behavior over products of curves.

Findings

Counterexample where tensor product of division algebras is not a domain

Analysis of Picard and Brauer groups of tensor products of fields

Splitting criterion for Brauer group elements over product of curves

Abstract

This paper began as an investigation of the question of whether $D_1 \otimes_F D_2$ 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 $F_1 \otimes_F F_2$ 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 $F$.