Counting Principles
Ian Stewart · 1990
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 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.