The ideal class group and the unit group

Abstract

Recall that the ideal class group of a number ring R consists of equivalence classes of nonzero ideals under the relation

$$ I\sim J\quad iff\quad \alpha I = \beta J\quad for{\text{ }}some{\text{ }}non{\text{ }}zero{\text{ }}\alpha ,\beta \in R; $$

the group operation is multiplication defined in the obvious way, and the fact that this is actually a group was proved in chapter 3 (Corollary 1 of Theorem 15). In this chapter we will prove that the ideal class group of a number ring is finite and establish some quantitative results that will enable us to determine the ideal class group in specific cases.