
Chapter 12
The Galois Correspondence
We are at last in a position to establish the fundamental properties of the Galois
correspondence between a field extension and its Galois group. Most of the work has
already been done, and all that remains is to put the pieces together.
12.1 The Fundamental Theorem of Galois Theory
Let us recall a few points of notation from Chapter 8. Let L : K be a field extension
in C with Galois group G, which consists of all K-automorphisms of L. Let F be the
set of intermediate fields, that is, subfields M such that K ⊆ M ⊆ L, and let G be the
set of all subgroups H of G. We have defined two maps
∗
: F → G
†
: G → F
as follows: if