Let $K$ be a number field, and let $\cal C$ be the set of ideal classes of $K$ with multiplication $\cdot$ defined by $$ [\mathfrak{a}] \cdot [\mathfrak{b}]=[\mathfrak{a} \mathfrak{b}] $$ where $\mathfrak{a}, \mathfrak{b}$ are ideals of ${\cal{O}}_K$

We shall check the group properties:

1. Associativity: $[\mathfrak{a}] \cdot ([\mathfrak{b}] \cdot [\mathfrak{c}])= [\mathfrak{a}] \cdot [\mathfrak{b}\mathfrak{c}]= [\mathfrak{a}(\mathfrak{b}\mathfrak{c})]= [\mathfrak{a}\mathfrak{b}\mathfrak{c}]= [(\mathfrak{a}\mathfrak{b})\mathfrak{c}]= [\mathfrak{a}\mathfrak{b}] \cdot [\mathfrak{c}]= ([\mathfrak{a}] \cdot [\mathfrak{b}]) \cdot [\mathfrak{c}]$
2. Identity element: $[ {\cal{O}}_K ] \cdot [\mathfrak{b}]= [\mathfrak{b}]=[\mathfrak{b}] \cdot [ {\cal{O}}_K ]$
3. Inverses: Consider $[\mathfrak{b}]$ Let $b$ be an integer in $\mathfrak{b}$ Then $\mathfrak{b} \supseteq (b)$ so there exists $\mathfrak{c}$ such that $\mathfrak{b}\mathfrak{c}=(b)$
   Then the ideal class $[\mathfrak{b}] \cdot [\mathfrak{c}] = [(b)]=[ {\cal{O}}_K ]$

It is abelian since $[\mathfrak{a}][\mathfrak{b}]=[\mathfrak{a}\mathfrak{b}]= [\mathfrak{b}\mathfrak{a}]=[\mathfrak{b}][\mathfrak{a}]$

This is group is called the ideal class group of $K$ The ideal class group is one of the principal objects of algebraic number theory. In particular, for an arbitrary number field $K$ very little is known about the size of this group, called the class number of $K$ See the analytic class number formula.