nLab
ideal class group

Definition

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

This was created to measure how far π’ͺ K\mathcal{O}_K is from being a PID.

More generally, for RR a Dedekind domain, we define C RC_R to be the group of fractional ideals of RR modded out by the group of principal ideals (freely generated as Frac(R)Frac(R)-modules). In other words, C RC_R := Pic(Spec 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 ΞΆ K\zeta_K of the number field KK has a simple pole at s=1s = 1. The class number formula says that its residue there is proportional the the product of the regulator with the class number of KK

limsβ†’1(sβˆ’1)ΞΆ K(s)∝ClassNumber Kβ‹…Regulator K. \underset{s\to 1}{\lim} (s-1) \zeta_K(s) \propto ClassNumber_K \cdot Regulator_K \,.

References

Last revised on May 11, 2018 at 10:33:14. See the history of this page for a list of all contributions to it.