Proposition 1

Let $K$ be a field, $L$ a $p$-radical extension of height $\leq 1$ of $K$ and $x = (x_i)_{i \in I}$, a family of elements of $L$. Then the vector space $K(x_i)_{i \in I}$ over $K$ is generated by the products $x'' = \prod_{i \in I} x_i^{\alpha_i}$ for $a$ in $A$. For $(x_i)_{i \in I}$ to be $p$-free (resp. a $p$-basis) it is necessary and sufficient that the family $(x^a)_{a \in A}$ should be free over $K$ (resp. a basis of $L$ over $K$).