nLab ideal class group

Definition

The ideal class group $C_K$ of a number field $K$ is defined as the group of fractional ideals of $\mathcal{O}_K$ modded out by the group of principal ideals in $\mathcal{O}_K$ (both groups freely generated as $K$-modules). An equivalent definition is that $C_K$ := $Pic(Spec \text{ } \mathcal{O}_K)$.

This was created to measure how far $\mathcal{O}_K$ is from being a PID.

More generally, for $R$ a Dedekind domain, we define $C_R$ to be the group of fractional ideals of $R$ modded out by the group of principal ideals (freely generated as $Frac(R)$-modules). In other words, $C_R$ := $Pic(Spec \text{ } R)$.

Properties

Relation to the idele class group

A much larger variant of the ideal class group is the idele class group…

Relation to the pole of the zeta function

The Dedekind zeta function $\zeta_K$ of the number field $K$ 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