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 $\rai{K}$ is a principal ideal of the ring extension $\rai{E}$ .
5. Every prime ideal ${\cal P}$ of $\rai{K}$ decomposes into the product of $\frac{h_K}{f}$ prime ideals in $\rai{E}$ , where $f$ is the order of $[{\cal P}]$ in the ideal class group of $\rai{E}$ .

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.