Galois Theory, 5th Edition

Chapter 10Counting Principles

DOI: 10.1201/9781003213949-10

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 $H\subseteq H^{†*}$, 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 that we need to restrict attention to finite extensions and ...