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 ${\mathcal{O}}_{K}$ is a principal ideal^{} of the ring extension ${\mathcal{O}}_{E}$.

5.
Every prime ideal^{} $\mathcal{P}$ of ${\mathcal{O}}_{K}$ decomposes into the product of $\frac{{h}_{K}}{f}$ prime ideals in ${\mathcal{O}}_{E}$, where $f$ is the order (http://planetmath.org/Order) of $[\mathcal{P}]$ in the ideal class group of ${\mathcal{O}}_{E}$.
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  20130322 12:36:45 
Last modified on  20130322 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 