Exercise 1, § 14

Let K be a commutative field, A a central simple K-algebra, L a semisimple commutative subalgebra of A, and $L'$ its commutant in A.

a) Suppose that A is the endomorphism algebra of a finite-dimensional K-vector space V. Prove that we have $[L : K][L': K]\geqslant [A : K]$ (cf. VIII, p. 78, Proposition 1); equality holds if and only if the L-module V is free. Deduce examples where the inequality is strict.

b) Prove the inequality $[L : K][L': K]\geqslant [A : K]$ in general.