existence of Hilbert class field

Let $K$ be a number field. There exists a finite extension $E$ of $K$ with the following properties:

1. $[E:K]=h_K$, where $h_K$ is the class number of $K$.

2. $E$ is Galois over $K$.

3. The ideal class group of $K$ is isomorphic to the Galois group of $E$ over $K$.

4. Every ideal of ${𝒪}_K$ is a principal ideal of the ring extension ${𝒪}_E$.

5. Every prime ideal $𝒫$ of ${𝒪}_K$ decomposes into the product of $\frac{h_K}{f}$ prime ideals in ${𝒪}_E$, where $f$ is the order (http://planetmath.org/Order) of $[𝒫]$ in the ideal class group of ${𝒪}_E$.

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

The field $E$ may also be characterized as the maximal abelian unramified (http://planetmath.org/AbelianExtension) extension 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 field.

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