Proposition 3

— Let $K$ be a field, $L$ a pure extension of $K$ and $(x_i)_{i \in I}$ a pure basis of $L$ (V, p. 106, Def. 2).

a) Let $V$ be a vector space over $L$, $A$ a derivation of $K$ into $V$ and $(v_i)_{i \in I}$ a family of elements of $V$. There exists a unique derivation $D$ of $L$ into $V$, extending $A$ and such that $D(x_i) = v_i$ for all $i \in I$.

b) The vector $L$-space $\Omega_K(L)$ of $K$-differentials of $L$ has the family $(dx_i)_{i \in I}$ as basis.

Assertion b) has been proved in IV, p. 23 and Assertion a) follows directly from this.