In this chapter we deal with the central results of Galois theory. The fundamental theorem on Galois theory establishes a one-to-one correspondence between the set of subfields of E, where E is a splitting field of a separable polynomial in F[x], and the set of subgroups of the group of F-automorphisms of E. This correspondence transforms certain problems about subfields of fields into more amenable problems about subgroups of groups. Among the applications, this serves as the basis of Galois's criterion for solvability of an equation by radicals, as discussed in the next chapter, and provides a simple algebraic proof of the fundamental theorem of algebra.

Automorphism groups and fixed fields

Let E be an extension of a field F. We denote by G(E/F) the group of automorphisms of E leaving each element of F fixed. The group G(E/F) is also called the group of F-automorphisms of E. Throughout this section we confine ourselves to finite separable extensions and their groups of automorphisms. We recall that a finite separable extension E of F is simple. Thus, E = F(α) for some α ∈ E. Let p(x) be the minimal polynomial of α over F. Then [F(α):F] = degree of p(x) = n, say. Also by Lemma 4.2, Chapter 15, we get that the order of the group G(E/F) is ≤ n. Thus, we have

1.1 Theorem.If E is a finite extension of a field F, then |G(E/F)| ≤ [E:F].