Division algebras, and more generally central simple algebras have played an important role in understanding field arithmetic. In 1944, Châtelet made an important observation, giving a correspondence between central simple algebras and a certain class of algebraic varieties, which became known as Severi-Brauer varieties. This observation has allowed for structural questions about central simple algebras to be recast as geometric problems about Severi-Brauer varieties, and vise-versa. In this talk, I'll describe some extensions of work of Karpenko along these lines, which relates the writing a central simple algebra as a tensor product of smaller algebras to questions about the Chow group of the corresponding Severi-Brauer variety.