# 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

$\left\{E\mid E\phantom{\rule{thickmathspace}{0ex}}\mathrm{is}\phantom{\rule{thickmathspace}{0ex}}a\phantom{\rule{thickmathspace}{0ex}}\mathrm{subfield}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}L\phantom{\rule{thickmathspace}{0ex}}\mathrm{containing}\phantom{\rule{thickmathspace}{0ex}}K\right\}\stackrel{\stackrel{\varphi }{\to }}{\underset{\psi }{←}}\left\{H\mid H\phantom{\rule{thickmathspace}{0ex}}\mathrm{is}\phantom{\rule{thickmathspace}{0ex}}a\phantom{\rule{thickmathspace}{0ex}}\mathrm{closed}\phantom{\rule{thickmathspace}{0ex}}\mathrm{subgroup}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}G\right\}$\{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

$\varphi \left(E\right)={\mathrm{Aut}}_{E}\left(L\right)$\phi(E) = Aut_E(L)

and

$\psi \left(H\right)={L}^{H}$\psi(H) = L^H

are bijective and inverse to each other. This correspondence reverses the inclusion relation: $K$ corresponds to $G$ and $L$ to $\left\{{\mathrm{id}}_{L}\right\}$.

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

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

$\left[E:K\right]\simeq \mathrm{index}\left[G:K\right]$ if $H$ is open;

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

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

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

$\mathrm{Gal}\left(E/K\right)\simeq G/H$ (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

Created on June 8, 2012 15:36:19 by Stephan Alexander Spahn (79.227.163.74)