Theorem 1

— Let K be a field of characteristic 0, L an extension of K and V a vector space over L. Let A be a derivation of K into V, $(x_i)_{i \in I}$, a transcendence basis of L over K and $(u_i)_{i \in I}$, a family of elements of V. Then there exists a unique derivation D of L into V extending A and such that $D(x_i) = u_i$ for all $i \in I$.

Put $E = K(x_i)_{i \in I}$; Prop. 3 (V, p. 128) shows that A extends to a unique derivation $D_0$ of E into V such that $D_0(x_i) = u_i$ for all $i \in I$. The field $L$ is an algebraic extension of E and since L is of characteristic 0, L is separable over E (V, p. 37, Cor.). Therefore (V, p. 129, Prop. 4) $D_0$ extends to a unique derivation D of $L$ into V.