The real numbers form a field. The complex numbers form a field extension. Notice that you get the complex numbers by adjoining a single root (i) of a polynomial p(x) = x² + 1, and then including everything you have to include because you have to make a field (e.g., 1+i, 2i, -i, etc). Notice, further, that the complex numbers have a single automorphism, which takes i to -i and -i to i (flip around the real axis), "permuting the roots."

Galois theory tells us that in general (starting with any field and almost any polynomial p(x)), one may create an extension field in which p(x) has a root, and that the extension field will have automorphisms moving the roots of p(x) to each other, but fixing the smaller field. Moreover, intermediate fields may exist, and if they do, they will be the parts not moved by certain subgroups of the group of automorphisms of the larger field. A beautiful correspondence can be established, then, between all the intermediate subfields and all the subgroups of the group of automorphisms of the larger field. I say "beautiful," because the subfields and subgroups match perfectly, with nothing left over on either side.

Because we can start with the field of integers modulo a prime, these ideas have powerful applications to number theory.