Proposition 15.3. Assuming that:

Then

(i) The set $I = {F ∕ K finite | F \subseteq L, F Galois}$ is a directed set under $\subseteq$ .
(ii) For $F, F^{'} \in I$ , $F \subseteq F^{'}$ there is a restriction map ${res}_{F, F^{'}} : Gal (F^{'} ∕ K) ↠ Gal (F ∕ K)$ and the natural map
$Gal (L ∕ K) \to \lim_{\overset{[}{F} \in I] \leftarrow} Gal (F ∕ K)$

is an isomorphism.