---
tag: KTKK9B1X41
kind: concept
name: Class number
msc: [11R29]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_class_group
---

## Summary

The class number $h_K$ of a [[number field|MWBDFS2KJ5]] $K$ is the order of its [[ideal class group|WQ46BD3ARY]].
It is always finite, and $h_K = 1$ exactly when the ring of integers $\mathcal{O}_K$ is a principal ideal domain, equivalently a unique factorisation domain.
Class numbers appear in the [[class number formula|55XX5A7XZV]], and the question of which fields have a given class number is the [[class number problem|KTGAS9DGRX]].

## Explanations

### L1 human

The class number counts how many essentially different kinds of ideals a number ring has.
If it is $1$, every ideal is generated by one number, and numbers factor into primes in only one way, as in the ordinary integers.
For example, the Gaussian integers $\Z[i]$ have class number $1$.
The ring $\Z[\sqrt{-5}]$ has class number $2$, and there $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ factors in two ways.