The Fundamental Theorem of Galois Theory

We saw in the last chapter that the Galois group of a finite extension of a field provides a lot of information about the structure of the extension field. In fact, if the extension is normal, then the degree of the extension is equal to the number of automorphisms belonging to the Galois group. In this chapter we encounter the Fundamental Theorem of Galois Theory, which shows that this connection between field extensions and groups carries even more information than that. In Chapter 49 we will be able to exploit this connection between field theory and group theory to address our goal of better understanding the solution of polynomial equations by field arithmetic and the extraction of roots.

Suppose that we have fields F ⊆ E ⊆ K. Then it is easy to see that Gal(K|E) is a subgroup of Gal(K|F ), because automorphisms of K that fix E clearly also fix F ⊆ E. Quick Exercise. Check that Gal(K|E) is not only a subset of Gal(K|F ) but also a subgroup.