Corollary 2

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

By Theorem 1, A is isomorphic to an algebra of the form $\prod^r_{i=1}\mathbf{M}_{n_i}(D_i)$ for an integer $r\geqslant 0$, integers $n_1, . . . , n_r$, and K-algebras of finite degree $D_1, . . . ,D_r$ that are fields and whose centers are separable extensions of K and therefore equal to K. By Proposition 3 of VIII, p. 231, we have $D_i= K$ for $i\in [1, r]$.