On the decomposition of a field as a tensor product

Abstract

The following two results in the theory of division algebras are well known and easily proved, for an arbitrary commutative field k (cf. for example [3, Chapter 10]).(i) The tensor product of two central division algebras over k of coprime degrees is again a division algebra.(ii) Every central division algebra over k is a tensor product of division algebras of prime power degrees.It is natural to ask whether corresponding results hold for commutative fields. The answers are not hard to find but (as far as I am aware) have not appeared in print before; since they throw some light on the nature of tensor products they seemed worth recording.