---
tag: RWK0HNF1NA
kind: concept
name: Regulator of a number field
msc: [11R27, 11R29]
sources:
  - url: https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem
---

## Summary

The regulator of an [[algebraic number field|MWBDFS2KJ5]] $K$ is a positive real number $R_K$ that measures the size of the unit group.
By [[Dirichlet's unit theorem|1D99CSKX6N]], the logarithms of the units form a lattice.
Up to normalisation, $R_K$ is the volume of a fundamental domain of that lattice.
It appears with the [[class number|KTKK9B1X41]] in the [[class number formula|55XX5A7XZV]] for the residue of the [[Dedekind zeta function|KXS5DQDFYD]].

## Explanations

### L1 human

The units of a number field are generated by a few fundamental units, up to roots of unity.
If the fundamental units are large numbers, the regulator is large.
For a real quadratic field with fundamental unit $\varepsilon > 1$, the regulator is $\log \varepsilon$.
For $\Q(\sqrt{2})$ the fundamental unit is $1 + \sqrt{2}$, so the regulator is $\log(1 + \sqrt{2}) \approx 0.881$.