State and prove fundamenta theorem of Galois theory
Answers
(The Fundamental Theorem of Galois Theory).
Let L/K be a finite Galois extension. Then there is an inclusion reversing
bijection between the subgroups of the Galois group Gal(L/K) and intermediary subfields L/M/K. Given a subgroup H, let M = L
H and
given an intermediary field L/M/K, let H = Gal(L/M).
Proof. This will be an easy consequence of all that has gone before.
Suppose that we are given a subgroup H of G. Let M = L
H and
then set K = Gal(L/M). We want to show that K = H. As we have
already proved that
H ⊂ K,
and |G| = [L : K] is finite, it suffices to prove that the cardinality of
K and is at most the cardinality of H. But But
|H| = [L : M],
and there are at most [L : M] automorphisms of L/M, so that
|K| ≤ [L : M] = |H|.
Thus H = K, and the composition one way is the identity.
Now suppose that we start with L/M/K. Let H = Gal(L/M) and
let N = L
H. We already know that
M ⊂ N,
and so by the Tower Law it suffices to prove that
[L : N] ≥ [L : M].
As L/K is Galois, then so is L/M. But then
[L : M] = |H|.
As H is a set of automorphisms of L/N, we have
[L : N] ≥ |H| = [L : M].
Thus M = N and the composition the other way is the identity. Thus
we have a correspondence. We have already seen that this correspondence is inclusion reversing.
The rest of the course will be adressed to deriving consequences of
the Fundamental Theorem. We start by observing