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main theorem of classical Galois theory

Main theorem of classical Galois theory

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

More precisely, the maps

{E|EisasubfieldofLcontainingK}ψϕ{H|HisaclosedsubgroupofG} \{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)

and

ψ(H)=L H \psi(H) = L^H

are bijective and inverse to each other. This correspondence reverses the inclusion relation: KK corresponds to GG and LL to {id L}\{id_L\}.

If EE corresponds to HH, then we have

  1. KEK \subset E is finite precisely if HH is open (in the profinite topology on GG)

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

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

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

  4. LEL \subset E is Galois precisely if HH is a normal subgroup of GG;

    Gal(E/K)G/HGal(E/K) \simeq G/H (as topological groups) if KEK \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 GG are its quotients by subgroups, which can be identified with transitive GG-sets. This naturally raises the question of what corresponds to non-transitive GG-sets.

category: Galois theory

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