ideal class

Let $K$ be a number field. Let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals in $\mathcal{O}_{K}$ (the ring of algebraic integers of $K$ ). Define a relation $\sim$ on the ideals of $\mathcal{O}_{K}$ in the following way: write $\mathfrak{a}\sim\mathfrak{b}$ if there exist nonzero elements $\alpha$ and $\beta$ of $\mathcal{O}_{K}$ such that $(\alpha)\mathfrak{a}=(\beta)\mathfrak{b}$ .

The relation $\sim$ is an equivalence relation, and the equivalence classes under $\sim$ are known as ideal classes.

The number of equivalence classes, denoted by $h$ or $h_{K}$ , is called the class number of $K$ .

Note that the set of ideals of any ring $R$ forms an abelian semigroup with the product of ideals as the semigroup operation. By replacing ideals by ideal classes, it is possible to define a group on the ideal classes of $\mathcal{O}_{K}$ in the following way.

Let $\mathfrak{a}$ , $\mathfrak{b}$ be ideals of $\mathcal{O}_{K}$ . Denote the ideal classes of which $\mathfrak{a}$ and $\mathfrak{b}$ are representatives by $[\mathfrak{a}]$ and $[\mathfrak{b}]$ respectively. Then define $\cdot$ by

$[\mathfrak{a}]\cdot[\mathfrak{b}]=[\mathfrak{a}\mathfrak{b}]$

Let ${\cal C}=\{[\mathfrak{a}]\mid\mathfrak{a}\neq(0),\mathfrak{a}\text{ an ideal % of }\mathcal{O}_{K}\}$ . With the above definition of multiplication, $\cal C$ is an abelian group, called the ideal class group (or frequently just the class group) of $K$ .

Note that the ideal class group of $K$ is simply the quotient group of the ideal group of $K$ by the subgroup of principal fractional ideals.

Title	ideal class
Canonical name	IdealClass
Date of creation	2013-03-22 12:36:42
Last modified on	2013-03-22 12:36:42
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	22
Author	mathcam (2727)
Entry type	Definition
Classification	msc 11R29
Synonym	ideal classes
Related topic	ExistenceOfHilbertClassField
Related topic	FractionalIdeal
Related topic	NumberField
Related topic	UnramifiedExtensionsAndClassNumberDivisibility
Related topic	ClassNumberDivisibilityInExtensions
Related topic	PushDownTheoremOnClassNumbers
Related topic	MinkowskisConstant
Related topic	ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivis
Defines	class number
Defines	ideal class group
Defines	class group