Corollary

— For each $x \in L - K$, there exists a p-basis of L over K containing X.

For $\{x\}$ is a p-free subset of $L$ over $K$ and so it is enough to apply Th. 2, a).

Let K be a field and $L$ an extension of K. If $D$ is a K-derivation of L with values in a vector L-space, then we have $D(x^p) = 0$ for all $x \in L$, and so $D$ is a $K(L^p)$-derivation; it follows that $\Omega_K(L) = \Omega_{K(L^p)}(L)$. Since $L$ is a p-radical extension of height $\leq 1$ of $K(L^p)$, we can apply the above results. For example, from Th. 1, c) we deduce the following: let $(x_i)_{i \in I}$ be a family of elements of $L$; for $(dx_i)_{i \in I}$ to be a basis of the vector L-space $\Omega_K(L)$ it is necessary and sufficient that $(x_i)_{i \in I}$ should be a p-basis of L over $K(L^p)$. Similarly, Cor. 2 of Prop. 5 implies: