Corollary 1

*Let $K$ be a field, $L$ a $p$-radical extension of height $\leq 1$ of $K$ and $V$ a vector space over $L$. Every derivation $A$ of $K$ into $V$ vanishing on $L^p$ extends to a derivation of $L$ into $V$.

By Zorn's lemma (E, III, p. 20) there exists a maximal extension of $A$ to a derivation $D_0 : L_0 \to V$, where $L_0$ is a subfield of $L$ containing $K$. Let $x \in L$; we have $x^p \in L_0$ and $A(x^p) = 0$, whence $D_0(x^p) = 0$. By Prop. 5,

D_0 extends to a derivation defined on L_0(x); by the maximal character of D_0 we thus have L_0(x) = L, whence x ∈ L_0. Since x was arbitrary we conclude that L_0 = L.