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单词 ExistenceOfHilbertClassField
释义

existence of Hilbert class field


Let K be a number fieldMathworldPlanetmath. There exists a finite extensionMathworldPlanetmath E of K with the following properties:

  1. 1.

    [E:K]=hK, where hK is the class numberMathworldPlanetmathPlanetmath of K.

  2. 2.

    E is Galois over K.

  3. 3.

    The ideal class group of K is isomorphicPlanetmathPlanetmathPlanetmath to the Galois groupMathworldPlanetmath ofE over K.

  4. 4.

    Every ideal of 𝒪K is a principal idealMathworldPlanetmathPlanetmath of the ring extension 𝒪E.

  5. 5.

    Every prime idealMathworldPlanetmathPlanetmath 𝒫 of 𝒪K decomposes into the product ofhKf prime ideals in 𝒪E, where f is the order (http://planetmath.org/Order)of [𝒫] in the ideal class group of 𝒪E.

There is a unique field E satisfying the above five properties, and it is known as the Hilbert class fieldMathworldPlanetmath of K.

The field E may also be characterized as the maximal abelianMathworldPlanetmath unramified (http://planetmath.org/AbelianExtension) extensionPlanetmathPlanetmath of K. Note that in this context, the term ‘unramified’ is meant not only for the finite places (the classical ideal theoretic ) but also for the infinite places. That is, every real embedding of K extends to a real embedding of E. As an example of why this is necessary, consider some real quadratic fieldMathworldPlanetmath.

Titleexistence of Hilbert class field
Canonical nameExistenceOfHilbertClassField
Date of creation2013-03-22 12:36:45
Last modified on2013-03-22 12:36:45
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id16
Authormathcam (2727)
Entry typeTheorem
Classificationmsc 11R32
Classificationmsc 11R29
Classificationmsc 11R37
Related topicIdealClass
Related topicGroup
Related topicNumberField
Related topicClassNumberDivisibilityInExtensions
Related topicRootDiscriminant
Related topicExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility
Related topicClassNumbersAndDiscriminantsTopicsOnClassGroups
DefinesHilbert class field
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更新时间:2025/5/4 19:23:07