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. The result is a bijection between subgroups of the Galois group and intermediate fields M. This bijection reverses inclusions: large subgroups correspond to smaller intermediate fields. It lets us study problems about fields using groups, and conversely.