# ideal classes form an abelian group

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. 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. 2.

Identity element: $[{\cal{O}}_{K}]\cdot[\mathfrak{b}]=[\mathfrak{b}]=[\mathfrak{b}]\cdot[{\cal{O}% }_{K}]$.

3. 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 $[\mathfrak{b}]\cdot[\mathfrak{c}]=[(b)]=[{\cal{O}}_{K}]$.

Then ${\cal C}$ is a group under the operation $\cdot$.

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.

 Title ideal classes form an abelian group Canonical name IdealClassesFormAnAbelianGroup Date of creation 2013-03-22 12:49:40 Last modified on 2013-03-22 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