Proposition 3

Let K be a separably closed field (V, §7, No. 8, p. 45, Definition 4), and let D be an algebra of finite degree over K that is a field. Then D is commutative.

Denote the characteristic exponent of K by $p$. Let $a$ be an element of D. The ring $K[a]$ is an algebraic extension of K (V, §3, No. 1, p. 17, Corollary 1). Since the field K is separably closed, it follows from V, §7, No. 7, p. 44, Proposition 13 that the algebra $K[a]$ is a $p$-radical extension of K. Hence there exists an integer $m\geqslant 0$ such that $a^{p^m}$ belongs to K. By Lemma 1, the field D is consequently commutative.