The Fundamental Theorem of Galois Theory states that:
Given a field extension that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group.
1) where and is the corresponding fixed field (the set of those elements in which are fixed by every automorphism in ).
2) where is an intermediate field of and is the set of those automorphisms in which fix every element of .
This correspondence is a one-to-one correspondence if and only if is a Galois extension.
Three Properties of the Galois Correspondence
- It is inclusing-reversing. The inclusion of subgroups holds iff the inclusion of fields holds.
- If is a subgroup of , then and .
- The field is a normal extension of (or equivalently, Galois extension, since any subextension of a separable extension is separable) iff is a normal subgroup of .