Exercise 6, § 8

Let A be a semisimple commutative algebra of finite rank over a commutative field K, and let $K_i$, for $i\in I$, be its simple factors, which are finite extensions of K. Let B be a subalgebra of A and $E_j(j\in J)$ its simple factors. Prove that there exist pairwise disjoint nonempty subsets $(\Lambda_j)_{j\in J}$ of I such that $E_j$ injects into $\prod_{i\in\Lambda_j}K_i$ for $j\in J$. For every $i\in \Lambda_j$, pr$_i$ induces an isomorphism from $E_j$ to a subfield $E_{ij}$ of $K_i$. Study the converse. Deduce that if each of the $K_i$ is separable over K, then A contains only finitely many subalgebras (cf. V, §6, No. 4, p. 30, Proposition 3).