Theorem 4

— Let N be a Galois extension of K and $\Gamma$ its Galois group. Let $\mathscr{K}$ be the set of subextensions of N and $\mathscr{G}$ the set of closed subgroups of $\Gamma$. For every subgroup $A \in \mathscr{G}$ we denote by $k(\Delta)$ the field of invariants of A and for every subfield $E \in \mathscr{K}$ we denote by $g(E)$ the group of E-*automorphisms* of N. Then $A \mapsto k(\Delta)$ is a bijection of $\mathscr{G}$ onto $\mathscr{K}$, and $E \mapsto g(E)$ is the inverse bijection.

A) The relation $E = k(g(E))$ (for $E \in \mathscr{K}$) is a consequence of the following more precise lemma: