ideal classes form an abelian group
Let $K$ be a number field, and let $\mathcal{C}$ be the set of ideal classes of $K$, with multiplication^{} $\cdot $ defined by
$$[\U0001d51e]\cdot [\U0001d51f]=[\U0001d51e\U0001d51f]$$ 
where $\U0001d51e,\U0001d51f$ are ideals of ${\mathcal{O}}_{K}$.
We shall check the group properties:

1.
Associativity: $[\U0001d51e]\cdot ([\U0001d51f]\cdot [\U0001d520])=[\U0001d51e]\cdot [\U0001d51f\U0001d520]=[\U0001d51e(\U0001d51f\U0001d520)]=[\U0001d51e\U0001d51f\U0001d520]=[(\U0001d51e\U0001d51f)\U0001d520]=[\U0001d51e\U0001d51f]\cdot [\U0001d520]=([\U0001d51e]\cdot [\U0001d51f])\cdot [\U0001d520]$

2.
Identity element^{}: $[{\mathcal{O}}_{K}]\cdot [\U0001d51f]=[\U0001d51f]=[\U0001d51f]\cdot [{\mathcal{O}}_{K}]$.
 3.
Then $\mathcal{C}$ is a group under the operation^{} $\cdot $.
It is abelian^{} since $[\U0001d51e][\U0001d51f]=[\U0001d51e\U0001d51f]=[\U0001d51f\U0001d51e]=[\U0001d51f][\U0001d51e]$.
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^{}.
Title  ideal classes form an abelian group 
Canonical name  IdealClassesFormAnAbelianGroup 
Date of creation  20130322 12:49:40 
Last modified on  20130322 12:49:40 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  13 
Author  mathcam (2727) 
Entry type  Theorem 
Classification  msc 11R04 
Classification  msc 11R29 
Related topic  NumberField 
Related topic  ClassNumbersAndDiscriminantsTopicsOnClassGroups 
Related topic  FractionalIdealOfCommutativeRing 
Defines  ideal class group 
Defines  class number 