Theorem 9.21.7 (Fundamental theorem of Galois theory). Let L/K be a finite Galois extension with Galois group G. Then we have K = L^ G and the map
\{ \text{subgroups of }G\} \longrightarrow \{ \text{subextensions }L/M/K\} ,\quad H \longmapsto L^ H
is a bijection whose inverse maps M to \text{Gal}(L/M). The normal subgroups H of G correspond exactly to those subextensions M with M/K Galois.
Proof.
By Lemma 9.21.4 given a subextension L/M/K the extension L/M is Galois. Of course L/M is also finite (Lemma 9.7.3). Thus |\text{Gal}(L/M)| = [L : M] by Lemma 9.21.2. Conversely, if H \subset G is a finite subgroup, then [L : L^ H] = |H| by Lemma 9.21.6. It follows formally from these two observations that we obtain a bijective correspondence as in the theorem.
If H \subset G is normal, then L^ H is fixed by the action of G and we obtain a canonical map G/H \to \text{Aut}(L^ H/K). This map has to be injective as \text{Gal}(L/L^ H) = H. Hence |G/H| = [L^ H : K] and L^ H is Galois by Lemma 9.21.2.
Conversely, assume that K \subset M \subset L with M/K Galois. By Lemma 9.15.7 we see that every element \tau \in \text{Gal}(L/K) induces an element \tau |_ M \in \text{Gal}(M/K). This induces a homomorphism of Galois groups \text{Gal}(L/K) \to \text{Gal}(M/K) whose kernel is H. Thus H is a normal subgroup.
\square
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