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 of E 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 of hKf 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.

Title existence of Hilbert class field
Canonical name ExistenceOfHilbertClassField
Date of creation 2013-03-22 12:36:45
Last modified on 2013-03-22 12:36:45
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Theorem
Classification msc 11R32
Classification msc 11R29
Classification msc 11R37
Related topic IdealClass
Related topic Group
Related topic NumberField
Related topic ClassNumberDivisibilityInExtensions
Related topic RootDiscriminant
Related topic ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility
Related topic ClassNumbersAndDiscriminantsTopicsOnClassGroups
Defines Hilbert class field