Corollary 2

—Let L be a p-radical extension of height ≤ l of a field K and let E be a subextension of L. Let U be the subspace of Ω_K(L) generated by the differentials of elements of E. Then E consists of the elements of L whose differential belongs to U. In particular we have d_{L/K}x ≠ 0 for all x ∈ L − K.

Let x be an element of L not belonging to E. Then {1, x, ..., x^{p^{l-1}}} is a basis of E(x) over E. Let A be the E-linear mapping of E(x) into L such that Δ(x^i) = ix' for 0 ≤ i < p. We verify at once that A is an E-derivation of E(x) into L, and in particular a K-derivation. By Cor. 1 to Prop. 5 there exists a K-derivation D of L into L extending A. By the universal property of Ω_K(L), there exists a linear form u on this vector L-space such that D = u ∘ d_{L/K}. We have D(x) = x and D|E = 0; therefore we have u(d_{L/K}x) ≠ 0 and u|U = 0, whence d_{L/K}x ∉ U.

The last assertion is the particular case E = K; then we have U = 0.