Theorem 1

—Let L be a p-radical extension of height ≤ l of a field K and (a_i)_{i ∈ I}, a family of elements of L.

a) For (x_i)_{i ∈ I} to be p-free over K it is necessary and sufficient that the family (dx_i)_{i ∈ I} should be free in the vector L-space Ω_K(L).

b) For (x_i)_{i ∈ I} to generate L over K it is necessary and sufficient that the family (dx_i)_{i ∈ I} should generate the vector L-space Ω_K(L).

c) For (x_i)_{i ∈ I} to be a p-basis of L over K it is necessary and sufficient that the family (dx_i)_{i ∈ I} should be a basis of the vector L-space Ω_K(L).

To begin with let us remark that the differential dx of an element of the field K((x_i)_{i ∈ I}) is a linear combination with coefficients in L of the differentials dx_i, i ∈ I. For the family (x_i)_{i ∈ I} to be p-free it is necessary and sufficient that x_i ∉ K(x_j)_{j ∈ I - {i}} for all i ∈ I (V, p. 99, Prop. 2). By Cor. 2 of Prop. 5 this means that dx_i is not a linear combination of the dx_j for j ≠ i in I. Assertion a) follows from this. Now b) follows at once from Cor. 2 to Prop. 5 and c) follows from a) and b).

The following corollary makes Cor. 1 of Prop. 5 more specific: