Theorem 1

— Let K be a field of characteristic 0. Every reduced K-algebra and in particular every extension of K is separable over K.

We first show that every extension L of K is separable. Let B be a transcendence basis of L over K (V, p. 109, Th. 1) and let $L_1 = K(B)$. Then $L_1$ is separable over K (V, p. 121, Prop. 6). Further, L is an algebraic extension of $L_1$ and $L_1$ is a field of characteristic 0; therefore L is separable over $L_1$ (V, p. 37, Cor.). By Prop. 9, L is thus separable over K.

Now let A be a reduced algebra over the field K. By the Cor. of Prop. 2 (V, p. 119) there exists a family $(L_i)_{i \in I}$ of extensions of K such that A is isomorphic to a subalgebra of $\prod_{i \in I} L_i$. Each of the algebras $L_i$ is separable over K by what has been said and so A has the same property, by Prop. 3 a) and c) (V, p. 119).