class number formula

Let K be a number fieldMathworldPlanetmath with [K:]=n=r1+2r2, where r1 denotes the number of real embeddings of K, and 2r2 is the number of complex embeddings of K. Let


be the Dedekind zeta function of K. Also define the following invariants:

  1. 1.

    hK is the class numberMathworldPlanetmathPlanetmath, the number of elements in the ideal class group of K.

  2. 2.

    RegK is the regulatorMathworldPlanetmath of K.

  3. 3.

    ωK is the number of roots of unityMathworldPlanetmath contained in K.

  4. 4.

    DK is the discriminantPlanetmathPlanetmathPlanetmath of the extensionPlanetmathPlanetmathPlanetmath K/.


Theorem 1 (Class Number Formula).

The Dedekind zeta function of K, ζK(s) converges absolutely for (s)>1 and extends to a meromorphic function defined for (s)>1-1n with only one simple poleMathworldPlanetmathPlanetmath at s=1. Moreover:


Note: This is the most general “class number formulaMathworldPlanetmath”. In particular cases, for example when K is a cyclotomic extension of , there are particular and more refined class number formulas.

Title class number formula
Canonical name ClassNumberFormula
Date of creation 2013-03-22 13:54:37
Last modified on 2013-03-22 13:54:37
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Theorem
Classification msc 11R29
Classification msc 11R42
Related topic FunctionalEquationOfTheRiemannZetaFunction
Related topic DedekindZetaFunction
Related topic IdealClass
Related topic Regulator
Related topic Discriminant
Related topic NumberField
Related topic ClassNumbersAndDiscriminantsTopicsOnClassGroups
Defines class number formula