If k is an algebraically closed field, and K|k and L|k are field extensions, then the tensor product of K and L over k is always a domain. It thus makes sense to conjecture that if D|K, E|L are central division algebras, then the tensor product of D and E over k is a (non-commutative) domain. We will show that this is often true, but not always. We will concentrate on the case that k has characteristic zero and E and D have prime index. We also hope to draw attention to the interesting properties of the tensor product of K and L and how they relate to our problem. Along the way we will make use of Picard varieties and elliptic curves.