---
tag: WN8PK6RQZR
kind: concept
name: Divisor class group
msc: [13C20, 13F05]
sources:
  - url: https://en.wikipedia.org/wiki/Fractional_ideal
---

## Summary

Let $A$ be a [[Krull domain|CB9WJCN9BM]].
Its divisor class group is the group of formal integer combinations of its height-one primes, modulo the divisors of nonzero fractions.
It is trivial exactly when the ring is a [[unique factorisation domain|06YMXJMXYS]], so it measures the failure of unique factorization.
For a Dedekind domain it is the ideal class group, and it contains the [[Picard group|DZW58R6T2Q]] of the ring.

## Explanations

### L1 human

In a ring with unique factorization, every height-one prime ideal is generated by one prime element.
In other rings some height-one primes are not principal, and the class group records which combinations of them are principal.
In $\Z[\sqrt{-5}]$, the prime $P = (2, 1 + \sqrt{-5})$ is not principal, but $P^2 = (2)$ is.
So the class of $P$ is a nonzero element of order $2$; in fact the class group of $\Z[\sqrt{-5}]$ has exactly two elements.