Remark

Let K be a field, L a p-radical extension of height $\leq 1$ of K and $(x_i)_i$, a family of elements of L. Let A be the polynomial algebra $K[(X_i)_{i \in I}]$, a the ideal of A generated by the polynomials $X_i^p - x_i^p$ and $\varphi : A \to L$ the K-algebra homomorphism such that $\varphi(X_i) = x_i$ for each $i \in I$. The family $(x_i)_i$, is p-free if and only if the kernel of $\varphi$ is equal to a. For $K[(X_i)]/a$ may be identified with the algebra $\bigotimes_{i \in I} K[X_i]/(X_i^p - x_i^p)$.

Let L be a p-radical extension of height $\leq 1$ of the field K. A subset S of $L$ is said to be p-free (resp. a p-basis) if the family defined by the identity mapping of S onto itself is p-free (resp. a p-basis). For a family $(x_i)_i$, of elements of $L$ to be p-free (resp. a p-basis) it is necessary and sufficient that the mapping $i \mapsto x_i$ should be a bijection of I on a p-free subset (resp. a p-basis) of L. By Prop. 1, every subset of a p-free set is p-free; conversely, if S is a subset of L all of whose finite subsets are p-free, then S is p-free. Finally, a p-basis of L over K is a p-free subset B such that $L = K(B)$.