
Chapter 10
Counting Principles
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.
It is largely for this reason ...