Exercise 17, § 12

Let K be a commutative field and D and E fields whose centers contain K. Suppose that $[E : K] =m$ is finite and that the center of one of the fields D and E is equal to K. The K-algebra $D\otimes_KE$ is then isomorphic to the endomorphism algebra of a finite-dimensional vector space $h$ over a field C (VIII, p. 222, Corollary 2).

a) Prove that $h$ divides $m$ (cf. VIII, p. 205, formula (32)). b) Let W be a vector space over D; then E is isomorphic to a K-subalgebra of End$_D(W)$ if and only if the dimension of W is infinite or a multiple of $m/h$ (note that W is then a $(D\otimes_K$ E)-module).

c) Prove that if [D : K] is finite and prime to $m$, then $D\otimes_KE$ is a field (apply a)).