Corollary 3

— Let L be an algebraic extension of K and M a commutative L-algebra (for example an extension of L). If M is separable over K, it is separable over L.

The algebra L is K-isomorphic to a subalgebra of M, hence L is a separable extension of K. Therefore (Cor. 2) there exists an L-isomorphism of $L^{p^{-\infty}}$ onto $K^{p^{-\infty}} \otimes_K L$. The ring $L^{p^{-\infty}} \otimes_L M$ is thus isomorphic to $(K^{p^{-\infty}} \otimes_K L) \otimes_L M$, and so to $K^{p^{-\infty}} \otimes_K M$ (II, p. 278, Prop. 2) and this latter ring is reduced because M is separable over K. Thus the ring $L^{p^{-\infty}} \otimes_L M$ is reduced, which proves that M is separable over L (V, p. 122, Th. 2).