---
tag: WQ46BD3ARY
kind: concept
name: Ideal class group
msc: [11R29, 13F05]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_class_group
same_as:
  wikidata: Q912083
  wikipedia: Ideal class group
---

## Summary

The ideal class group of a [[Dedekind domain|7NPY2CN5KB]] is the group of nonzero fractional ideals modulo the principal ones.
The main case is the ring of integers of a [[number field|MWBDFS2KJ5]].
It is trivial exactly when the ring is a [[principal ideal domain|0CMD0BCH8T]].
For a Dedekind domain this is the same as being a [[unique factorisation domain|0W4X9G6263]].
So it measures how far unique factorisation of elements fails.
For a number field it is a finite abelian group, and its order is the [[class number|KTKK9B1X41]].

## Explanations

### L1 human

In a number ring, numbers may factor in several ways, but ideals factor uniquely.
Some ideals are generated by a single number, and those behave just like numbers; others are not.
The class group lumps together ideals that differ by a principal ideal and records how they multiply.
For $\Z[\sqrt{-5}]$ there are two classes: the principal ideals, and the class of the ideal $(2, 1 + \sqrt{-5})$, whose square is the principal ideal $(2)$.