Theorem 3

— Let K be a perfect field. a) Every algebraic extension of K is a perfect field. b) Every extension of K is separable. c) For a K-algebra to be separable it is necessary and sufficient that it should be reduced. d) Let A and B be two reduced K-algebras. Then $A \otimes_K B$ is reduced. e) If E and F are two extensions of K, then the ring $E \otimes_K F$ is reduced.

Assertion a) is just Cor. 1 of Prop. 11 (V, p. 43). Assertion b) follows from Th. 1 (V, p. 122) when K is of characteristic 0 and from Cor. 1 (V, p. 123) when K is of characteristic $p \neq 0$. Let us prove c). The case where K is of characteristic 0 follows from Th. 1 (V, p. 122). So it is enough to show that if K is perfect of characteristic $p \neq 0$ and A is a reduced K-algebra, then A is separable over K. But this follows from the equivalence of the conditions a) and b) of Th. 2 (V, p. 122; take $K' = K$ in b)). Finally, d) follows from c) and Prop. 5 (V, p. 120) and e) is a particular case of d).