Hilbert class field

Summary

The Hilbert class field of a number field $K$ is the largest abelian extension of $K$ that is unramified at every place. By class field theory, its Galois group over $K$ is isomorphic to the ideal class group of $K$, so its degree is the class number. A prime ideal of $K$ splits completely in it exactly when it is principal. So the class group, an arithmetic object, is realised as a Galois group.