nLab main theorem of classical Galois theory

Main theorem of classical Galois theory

Let $K \subset L$ be a Galois extension of fields with Galois group $G$. Then the intermediate fields of $K \subset L$ correspond bijectively to the closed subgroups of $G$.

More precisely, the maps

$$\{E | E\;is\;a\;subfield\;of\;L\;containing\;K\} \stackrel{\overset{\phi}{\to}}{\underset{\psi}{\leftarrow}} \{H|H\;is\;a\;closed\;subgroup\;of\;G\}$$

defined by

$$\phi(E) = Aut_E(L)$$

and

$$\psi(H) = L^H$$

are bijective and inverse to each other. This correspondence reverses the inclusion relation: $K$ corresponds to $G$ and $L$ to $\{id_L\}$.

If $E$ corresponds to $H$, then we have

1. $K \subset E$ is finite precisely if $H$ is open (in the profinite topology on $G$)

   $[E:K] \simeq index[G:K]$ if $H$ is open;

2. $E \subset L$ is Galois with $Gal(L/E) \simeq H$ (as topological groups);

3. for every $\sigma \in G$ we have that $\sigma[E]$ corresponds to $\sigma H \sigma^{-1}$;

4. $L \subset E$ is Galois precisely if $H$ is a normal subgroup of $G$;

   $Gal(E/K) \simeq G/H$ (as topological groups) if $K \subset E$ is Galois.

This appears for instance as Lenstra, theorem 2.3.

This suggests that more fundamental than the subgroups of a Galois group $G$ are its quotients by subgroups, which can be identified with transitive $G$-sets. This naturally raises the question of what corresponds to non-transitive $G$-sets.

category: Galois theory