Theorem 3

— *Let K be a field and L an extension of K. For L to be separable over K it is necessary and sufficient that every derivation of K into a vector L-space should extend to a derivation of L. We may assume that K has characteristic $\neq 0$. Suppose first that L is separable over K. Let $V$ be a vector space over L and $A$ a derivation of K into $V$. By Mac

Lane's criterion (V, p. 124, Remark), the fields $L^p$ and $K$ are linearly disjoint over $K^p$. Since $A$ is a $K^p$-linear mapping of $K$ into $V$, it extends in a unique way to an $L^p$-linear mapping $A'$ of $K[L^p] = K(L^p)$ into $V$. It is clear that $A'$ is a derivation of $K(L^p)$ into $V$ which vanishes on $L^p$; thus it extends (V, p. 103, Cor. 3) to a derivation of $L$ into $V$.

Conversely, suppose that every derivation of $K$ with values in $L$ extends to a derivation of $L$ into $L$. Let $B$ be a p-basis of $K$ (V, p. 99) and let $A$ be the set of families $(\alpha_h)_{h \in R}$ with finite support consisting of integers between $()$ and $p - 1$. For each $b \in B$ there exists a derivation $\Delta_b$ of $K$ into $K$ characterized by $\Delta_b(b') = \delta_{bb'}$ (Kronecker symbol) for each $b' \in B$ (V, p. 103). By hypothesis there exists for each $b \in B$ a derivation $D_b$ of $L$ into $L$ extending $\Delta_b$. We have $D_b(b') = \delta_{bb'}$ for $b, b'$ in $B$, which proves that in $\Omega_{1/p}(L)$ the differentials $db$ (for $b \in B$) are linearly independent over $L$. Therefore (V, p 102, Th. 1), $B$ is p-free over $L^p$. It follows (V, p. 98, Prop. 1 and p. 124, Remark) that the extension $L$ of $K$ is separable.