Finiteness of the Class Group

Frequently $\O_K$ is not a principal ideal domain. This chapter is about a way to understand how badly $\O_K$ fails to be a principal ideal domain. The class group of $\O_K$ measures this failure. As one sees in a course on Class Field Theory, the class group and its generalizations also yield deep insight into the extensions of $K$ that are Galois with abelian Galois group.