Definition 1

Let $K$ be a field and $L$ a $p$-radical extension of height $\leq 1$ of $K$. A family $(x_i)_{i \in I}$ of elements of $L$ is said to be $p$-free over $K$ (resp. a $p$-basis of $L$ over $K$) if $x_i \notin K$ for all $i \in I$ and the homomorphism of $\bigotimes_{i \in I} K(x_i)$ into $L$ derived from the canonical injections $u_i : K(x_i) \to L$ is injective (resp. bijective).

If $a$ is an element of $L - K$, it is $p$-radical of height 1, and so its minimal polynomial over $K$ is $X^p - a^p$ ($V$, p. 24, Prop. 1); therefore $\{1, a, \ldots, a^{p-1}\}$ is a basis of $K(a)$ over $K$. Let $(x_i)_{i \in I}$ be a family of elements of $L - K$ and $A$ the subset of $\mathbf{N}^{(I)}$ consisting of all families of finite support $a = (\alpha_i)_{i \in I}$, such that $\alpha_i < p$ for all $i \in I$; Prop. 9 (III, p. 471) shows that the elements $\bigotimes_{i \in I} x_i^{\alpha_i}$ as $a$ runs over $A$ form a basis of the vector space $\bigotimes_{i \in I} K(x_i)$ over $K$. Moreover, the canonical homomorphism of $\bigotimes_{i \in I} K(x_i)$ into $L$ has as image $K(x_i)_{i \in I}$ ($V$, p. 18, Cor. 1), we thus have the following proposition: