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 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 Hilbert class field of $K$.

The field $E$ may also be characterized as the maximal abelian unramified extension of $K$. Note that in this context, the term ‘unramified’ is meant not only for the finite places (the classical ideal theoretic interpretation) 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.