Corollary

—Let (x_i)_{i ∈ I} be a p-basis of L over K. Let V be a vector space over L, A a derivation of K into V which vanishes on L^p and (u_i)_{i ∈ I} a family of elements of V. Then there exists a unique derivation D of L in V which extends A and maps x_i to u_i for all i ∈ I.

By Cor. 1 of Prop. 5 there exists a derivation D_0 of L into V extending A. The derivations of L in V extending A are precisely the mappings of the form D = D_0 + u ∘ d_{L/K} where u is an L-linear mapping of Ω_K(L) into V. We have D(x_i) = u_i if and only if the linear mapping u satisfies the conditions u(dx_i) = u_i − D_0(x_i). Since the family dx_i is a basis of Ω_K(L), this determines u uniquely and the corollary follows.

Let L be a p-radical extension of height $\leq 1$ of K and $(x_i)_{i \in I}$ a p-basis of L over K. By the Cor. to Th. 1 there exists for each $i \in I$ a K-derivation $D_i$ of L into L characterized by $D_i(x_j) = \delta_{ij}$ (Kronecker symbol); $D_i$ is sometimes called the partial derivation with respect to $x_i$. When $I$ is finite, the family $(D_i)_{i \in I}$ is a basis of the vector space over L consisting of the K-derivations of L into L.

Th. 1 allows the study of p-bases to be reduced to the study of bases of a vector space. For example we have the following result: