ideal classes form an abelian group

Let K be a number field, and let 𝒞 be the set of ideal classes of K, with multiplicationPlanetmathPlanetmath defined by


where 𝔞,𝔟 are ideals of 𝒪K.

We shall check the group properties:

  1. 1.

    Associativity: [𝔞]([𝔟][𝔠])=[𝔞][𝔟𝔠]=[𝔞(𝔟𝔠)]=[𝔞𝔟𝔠]=[(𝔞𝔟)𝔠]=[𝔞𝔟][𝔠]=([𝔞][𝔟])[𝔠]

  2. 2.

    Identity elementMathworldPlanetmath: [𝒪K][𝔟]=[𝔟]=[𝔟][𝒪K].

  3. 3.

    InversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath: Consider [𝔟]. Let b be an integer in 𝔟. Then 𝔟(b), so there exists 𝔠 such that 𝔟𝔠=(b).
    Then the ideal class [𝔟][𝔠]=[(b)]=[𝒪K].

Then 𝒞 is a group under the operationMathworldPlanetmath .

It is abelianMathworldPlanetmath since [𝔞][𝔟]=[𝔞𝔟]=[𝔟𝔞]=[𝔟][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 theoryMathworldPlanetmath. 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 formulaMathworldPlanetmath.

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