Remark

— Mac Lane's criterion may be formulated without introducing any extensions of K other than L. For by c) of Cor. 1, L is separable over K if and only if L and $K^P$ are linearly disjoint over K. Since the mapping $x \mapsto x^p$ is an isomorphism of L onto the subfield $L^P$, we obtain the following criterion (cf. V, p. 177, Ex. 11 for an analogous criterion for algebras):

L is separable over K if and only if the subfields $L^P$ and K of L are linearly disjoint over $K^P$.