main theorem of classical Galois theory

Main theorem of classical Galois theory

Let KL be a Galois extension of fields with Galois group G. Then the intermediate fields of KL correspond bijectively to the closed subgroups of G.

More precisely, the maps

{EEisasubfieldofLcontainingK}ψϕ{HHisaclosedsubgroupofG}\{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

ϕ(E)=Aut E(L)\phi(E) = Aut_E(L)


ψ(H)=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 {id L}.

If E corresponds to H, then we have

  1. KE is finite precisely if H is open (in the profinite topology on G)

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

  2. EL is Galois with Gal(L/E)H (as topological groups);

  3. for every σG we have that σ[E] corresponds to σHσ 1;

  4. LE is Galois precisely if H is a normal subgroup of G;

    Gal(E/K)G/H (as topological groups) if KE 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 (