Corollary

— If $L$ is a separable extension of $K$, the canonical mapping $\alpha_p : \Omega_p(K) \otimes_K L \to \Omega_p(L)$ is injective for every subfield $P$ of $K$. Conversely, if there exists a perfect subfield $P$ of $K$ (for example the prime subfield of $K$) such that the mapping $\alpha_p$ is injective, then $L$ is separable over $K$.

The first assertion follows from Prop. 2 (V, p. 128) and Th. 3. Conversely, let $P$ be a perfect subfield of $K$; we have $P = P^p \subset K^p$, hence every derivation of $K$ into a vector $L$-space is a $P$-derivation; the second assertion of the corollary now follows from Cor. 2 (V, p. 130) and Th. 3.