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

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