ideal class

Let K be a number field. Let π”ž and π”Ÿ be ideals in π’ͺK (the ring of algebraic integers of K). Define a relationMathworldPlanetmathPlanetmath ∼ on the ideals of π’ͺK in the following way: write π”žβˆΌπ”Ÿ if there exist nonzero elements Ξ± and Ξ² of π’ͺK such that (Ξ±)β’π”ž=(Ξ²)β’π”Ÿ.

The relation ∼ is an equivalence relationMathworldPlanetmath, and the equivalence classesMathworldPlanetmath under ∼ are known as ideal classesMathworldPlanetmath.

The number of equivalence classes, denoted by h or hK, is called the class numberMathworldPlanetmath of K.

Note that the set of ideals of any ring R forms an abelian semigroup with the product of ideals as the semigroup operationMathworldPlanetmath. By replacing ideals by ideal classes, it is possible to define a group on the ideal classes of π’ͺK in the following way.

Let π”ž, π”Ÿ be ideals of π’ͺK. Denote the ideal classes of which π”ž and π”Ÿ are representatives by [π”ž] and [π”Ÿ] respectively. Then define β‹… by


Let π’ž={[π”ž]βˆ£π”žβ‰ (0),π”žβ’Β an ideal of ⁒π’ͺK}. With the above definition of multiplicationPlanetmathPlanetmath, π’ž is an abelian groupMathworldPlanetmath, 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 groupMathworldPlanetmath of the ideal group of K by the subgroupMathworldPlanetmathPlanetmath 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