ABSTRACT
The basic idea behind Galois theory is that given a field extension L/K, there is a group G consisting of automorphisms of L that fix K (called K-automorphisms). Every subgroup H of G defines a field H† lying between K and L, and every such ‘intermediate’ field M defines a subgroup M* of G. When proving the Fundamental Theorem of Galois Theory in Chapter 12, we need to show that if H is a subgroup of the Galois group of a finite normal extension L/K, then H†∗ = H. Our method is to show that H and H†∗ are finite groups and have the same order. Since we already know that H ⊆ H†∗, the two groups must be equal. This is an archetypal application of a counting principle: showing that two finite sets, one contained in the other, are identical, by counting how many elements they have and showing that the two numbers are the same.
