Ideal class group

Summary

The ideal class group of a Dedekind domain is the group of nonzero fractional ideals modulo the principal ones. The main case is the ring of integers of a number field. It is trivial exactly when the ring is a principal ideal domain. For a Dedekind domain this is the same as being a unique factorisation domain. So it measures how far unique factorisation of elements fails. For a number field it is a finite abelian group, and its order is the class number.