nLab class number

Definition

In number theory, the class number of a number field is the order of its ideal class group.

Properties

Relation to the pole of the zeta function

The Dedekind zeta function $\zeta_K$ of $K$ has a simple pole at $s = 1$. The class number formula says that its residue there is proportional the the product of the regulator with the class number of $K$

$$\underset{s\to 1}{\lim} (s-1) \zeta_K(s) \propto ClassNumber_K \cdot Regulator_K \,.$$

References

• Wikipedia, Ideal class group