Theorem 5

*a) The mapping $A \mapsto K(\mathscr{P}^{-1}(A))$ is a bijection of the set of subgroups of $K$ containing $\mathscr{P}(K)$ onto the set of abelian subextensions of exponent dividing $p$ in $\Omega$. The inverse mapping is $L \mapsto \mathscr{P}(L) \cap K$. b) For every subgroup $A$ of $K$ containing $\mathscr{P}(K)$, the homomorphism

is bijective, and it is a homeomorphism when $\mathrm{Hom}(A/\mathscr{P}(K), \mathbf{F}_p)$ is equipped with the topology of simple convergence. c) Let $A$ be a subgroup of $K$ containing $\mathscr{P}(K)$ and let $B$ be a basis of the vector $\mathbf{F}_p$-space $A/\mathscr{P}(K)$. For each $a \in B$ let $x_a$ be an element of $\Omega$ such that $\mathscr{P}(x_a)$ is a representative of $a$ in $A$. Then the monomials $x^\alpha = \prod_{a \in B} x_a^{\alpha(a)}$ with a = (\alpha(a)) in N^{(B)} such that 0 \leq \alpha(a) < p for each a \in B form a basis of the vector K-space K(\mathfrak{p}^{-1}(A)). In particular we have [K(\mathfrak{p}^{-1}(A)):K] = (A:\mathfrak{p}(K)).

Th. 5 is proved in the same way as Th. 4 (V, p. 89) mutatis mutandis.