Proof
The proof of the fundamental theorem is not trivial. The crux in the usual treatment is a rather delicate result of Emil Artin which allows one to control the dimension of the intermediate field fixed by a given group of automorphisms. The automorphisms of a Galois extension K/F are linearly independent as functions over the field K. The proof of this fact follows from a more general notion, namely, the linear independence of characters.
There is also a fairly simple proof using the primitive element theorem. This proof seems to be ignored by most modern treatments, possibly because it requires a separate (but easier) proof in the case of finite fields.
In terms of its abstract structure, there is a Galois connection; most of its properties are fairly formal, but the actual isomorphism of the posets requires some work.
Read more about this topic: Fundamental Theorem Of Galois Theory
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