Let be the set of integers in an algebraic number field , and let denote the set of equivalence classes of ideals in under the relation if and only if for some nonzero . Prove that is an abelian group under the usual ideal product of class representatives.

We will denote by the equivalence class containing the ideal . We showed in this previous exercise that is a well-defined binary operator on .

For all , we have , so that our product is associative.

For all , we have , and similarly .

For all , there exists (by Theorem 8.13) an ideal such that is principal. Thus , and so every element of has an inverse.

Hence is a group.

Moreover, we have , so that is abelian.