ABSTRACT
When proving the Fundamental Theorem of Galois theory in Chapter 12, we will need to show that if H is a subgroup of the Galois group of a finite normal extension L : K, then H†∗ =H. Here the maps ∗ and † are as defined in Section 8.6. Our method will be 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.
