Corollary

Let K be a separably closed field and A a semisimple algebra of finite degree over K. Then there exist an integer $r\geqslant 0$, strictly positive integers $n_1, . . . , n_r$, and extensions $K_1, . . . ,K_r$ of K of finite degree such that A is isomorphic to the algebra $\prod_{i=1}^r\mathbf{M}_{n_i}(K_i)$.

By the structure theorem for semisimple algebras (VIII, p. 135, Theorem 1), A is isomorphic to an algebra $\prod^r_{i=1}\mathbf{M}_{n_i}(D_i)$, where $r$ is an integer $\geqslant 0,n_1, . . . , n_r$ are strictly positive integers, and $D_1, . . . ,D_r$ are K-algebras of finite degree that are fields. Since the field K is separably closed, each field $D_i$ is commutative by Proposition 3; the corollary follows.