---
tag: 1TQ2F9CFVE
kind: statement
role: theorem
name: Minkowski bound
msc: [11R29, 11H06]
uses: [MWBDFS2KJ5, E0VZTNPJXA, WQ46BD3ARY, M54ADM0CVF, 0K387S2PKQ, WN9V7WSJKF, KTKK9B1X41, 1D99CSKX6N]
about: [WQ46BD3ARY, E0VZTNPJXA]
sources:
  - url: https://en.wikipedia.org/wiki/Minkowski%27s_bound
same_as:
  wikidata: Q6867845
  wikipedia: Minkowski's bound
---

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] of degree $n$, with ring of integers $\mathcal{O}_K$ and [[discriminant|E0VZTNPJXA]] $d_K$.
Let $r_2$ be the number of pairs of complex conjugate embeddings $K \to \C$.
Put
$$M_K = \sqrt{\abs{d_K}} \left(\frac{4}{\pi}\right)^{r_2} \frac{n!}{n^n}.$$
Then every class of the [[ideal class group|WQ46BD3ARY]] of $K$ contains a nonzero ideal $\mathfrak{a} \subseteq \mathcal{O}_K$ whose [[norm|M54ADM0CVF]] satisfies $N(\mathfrak{a}) \le M_K$.

## Remarks

The proof embeds $\mathcal{O}_K$ and its ideals as [[lattices|WN9V7WSJKF]] in $\R^n$.
[[Minkowski's theorem|0K387S2PKQ]] on convex bodies then gives, in each nonzero ideal $\mathfrak{b}$, a nonzero element $\alpha$ with $\abs{N_{K/\Q}(\alpha)} \le M_K N(\mathfrak{b})$.
Applied to an ideal $\mathfrak{b}$ in the inverse class, the ideal $\alpha \mathfrak{b}^{-1}$ is integral, lies in the given class and has norm at most $M_K$.

Consequence: the class group is generated by the classes of the prime ideals of norm at most $M_K$.
Since there are only finitely many ideals of bounded norm, the class group is finite.

Example: for $K = \Q(\sqrt{-5})$ one has $n = 2$, $r_2 = 1$, $d_K = -20$, so $M_K = \sqrt{20} \cdot \frac{4}{\pi} \cdot \frac{2}{4} \approx 2.85$.
Only the prime ideal $\mathfrak{p} = (2, 1 + \sqrt{-5})$ of norm $2$ needs checking.
It is not principal, since $a^2 + 5b^2 = 2$ has no integer solution, and $\mathfrak{p}^2 = (2)$; so the class number is $2$.
For $K = \Q(i)$, $M_K = 2 \cdot \frac{4}{\pi} \cdot \frac{2}{4} \approx 1.27 < 2$, so every class contains the ideal $(1)$ and $\Z[i]$ is a principal ideal domain.

## Explanations

### L1 human

The class group of a number field measures how far its integers are from having unique factorisation.
The Minkowski bound is an explicit number such that only the prime ideals below it matter.
So the class group of a given field can be computed by checking finitely many small primes.

### L2 human

The bound is the standard tool for computing [[class numbers|KTKK9B1X41]] by hand for fields of small degree.
Since every class contains an integral ideal of norm at least $1$, it also gives $M_K \ge 1$, which yields a lower bound for $\abs{d_K}$; this is the [[Hermite–Minkowski theorem|KMVW45B0TT]].
The same geometry of numbers proves [[Dirichlet's unit theorem|1D99CSKX6N]].