Theorem 2

— Let L be a p-radical extension of height $\leq 1$ of a field K. a) There exist p-bases of L over K. More precisely, if S is a p-free subset of L over K and T a subset of L such that $S \subset T$ and $L = K(T)$, then there exists a p-basis B of L over K such that $S \subset B \subset T$. b) Two p-bases of L over K have the same cardinal. c) For $[L:K]$ to be finite it is necessary and sufficient that the vector space $\Omega_K(L)$ over L should be of finite dimension over L and then we have

Assertions a) and b) are immediate (II, p. 292). If $[L : K]$ is finite, there exists by a) a finite p-basis $(x_1, ..., x_n)$ of $L$ over $K$; then the monomials $x_1^{\alpha_1} ... x_n^{\alpha_n}$ with $0 \leq \alpha_i < p$ for $1 \leq i \leq n$ form a basis of $L$ over $K$ and the differentials $dx_1, ..., dx_n$ form a basis of $\Omega_K(L)$ over $L$. We thus have $[L : K] = p^n$ and $[\Omega_K(L) : L] = n$. Conversely if $\Omega_K(L)$ is of finite dimension over $L$, there exists a finite p-basis of L over K (V, p. 102, Th. 1) and $[L : K]$ is finite.