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单词 ClassNumberFormula
释义

class number formula


Let K be a number fieldMathworldPlanetmath with [K:]=n=r1+2r2, where r1denotes the number of real embeddings of K, and 2r2 is thenumber of complex embeddings of K. Let

ζK(s)

be the Dedekind zeta function of K. Also define the followinginvariants:

  1. 1.

    hK is the class numberMathworldPlanetmathPlanetmath, the number of elements in theideal 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/.

Then:

Theorem 1 (Class Number Formula).

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

lims1(s-1)ζK(s)=2r1(2π)r2hKRegKωKDK

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

Titleclass number formula
Canonical nameClassNumberFormula
Date of creation2013-03-22 13:54:37
Last modified on2013-03-22 13:54:37
Owneralozano (2414)
Last modified byalozano (2414)
Numerical id5
Authoralozano (2414)
Entry typeTheorem
Classificationmsc 11R29
Classificationmsc 11R42
Related topicFunctionalEquationOfTheRiemannZetaFunction
Related topicDedekindZetaFunction
Related topicIdealClass
Related topicRegulator
Related topicDiscriminant
Related topicNumberField
Related topicClassNumbersAndDiscriminantsTopicsOnClassGroups
Definesclass number formula
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