# existence of Hilbert class field

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

1. 1.

$[E:K]=h_{K}$, where $h_{K}$ is the class number of $K$.

2. 2.

$E$ is Galois over $K$.

3. 3.

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

4. 4.

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

5. 5.

Every prime ideal ${\cal P}$ 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 $[{\cal P}]$ 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 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