---
tag: F99RM1T02C
kind: statement
role: theorem
name: Finiteness of the class number
msc: [11R29, 11R04]
uses: [MWBDFS2KJ5, WQ46BD3ARY, M54ADM0CVF, 1TQ2F9CFVE, E0VZTNPJXA, 0K387S2PKQ, 7NPY2CN5KB, 1D99CSKX6N, 55XX5A7XZV, KXS5DQDFYD]
about: [WQ46BD3ARY, KTKK9B1X41]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_class_group
---

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]].
Then the [[ideal class group|WQ46BD3ARY]] $\operatorname{Cl}(K)$ is finite.

## Remarks

The standard proof uses the [[Minkowski bound|1TQ2F9CFVE]].
Let $n = [K : \Q]$, let $2r_2$ be the number of non-real embeddings of $K$ into $\C$, and let $d_K$ be the [[discriminant|E0VZTNPJXA]] of $K$.
By [[Minkowski's theorem|0K387S2PKQ]] on lattice points, every ideal class contains an integral ideal $\mathfrak{a}$ with [[norm|M54ADM0CVF]]
$$N(\mathfrak{a}) \le \frac{n!}{n^n} \left(\frac{4}{\pi}\right)^{r_2} \sqrt{\abs{d_K}}.$$
There are only finitely many ideals of $\mathcal{O}_K$ of bounded norm, so there are finitely many classes.
Example: for $K = \Q(\sqrt{-5})$, $n = 2$, $r_2 = 1$ and $d_K = -20$, so the bound is $\tfrac12 \cdot \tfrac4\pi \cdot \sqrt{20} \approx 2.85$; the ideals of norm at most $2$ are $\mathcal{O}_K$ and $(2, 1 + \sqrt{-5})$, which gives $h_K \le 2$.
The hypothesis that $K$ is a number field is needed: for a general [[Dedekind domain|7NPY2CN5KB]] the class group can be infinite.

## Explanations

### L1 human

In a number ring, unique factorisation can fail, and the class group measures by how much.
This theorem says that the failure is always finite: there are only finitely many kinds of ideals up to multiplication by numbers.
It turns the question of unique factorisation in a given field into a finite computation.

### L2 human

The theorem is one of the two basic finiteness results of algebraic number theory, the other being [[Dirichlet's unit theorem|1D99CSKX6N]].
Both come from the geometry of numbers: $\mathcal{O}_K$ embeds as a lattice in $\R^n$, and Minkowski's convex body theorem produces small elements.
The explicit bound makes class groups computable; for example it shows at once that $\Q(i)$, $\Q(\sqrt{-2})$ and $\Q(\sqrt{-3})$ have class number $1$.
The size of $h_K$ is described by the [[class number formula|55XX5A7XZV]], which links it to the residue of the [[Dedekind zeta function|KXS5DQDFYD]] at $s = 1$.