---
tag: 3AG1JV6J80
kind: definition
name: Regulator of a number field
msc: [11R27, 11R29]
defines: RWK0HNF1NA
uses: [MWBDFS2KJ5, 1D99CSKX6N, 00KDZRW4C0, 04DXAFN2ES]
sources:
  - url: https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem
---

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\mathcal{O}_K$.
Let $\sigma_1, \ldots, \sigma_{r_1}$ be the real embeddings of $K$, and let $\sigma_{r_1 + 1}, \ldots, \sigma_{r_1 + r_2}$ be one embedding from each pair of complex conjugate non-real embeddings.
Put $N_j = 1$ for $j \le r_1$ and $N_j = 2$ for $j > r_1$, and let $r = r_1 + r_2 - 1$.
Let $u_1, \ldots, u_r$ be a system of fundamental units, as in [[Dirichlet's unit theorem|1D99CSKX6N]].
If $r \ge 1$, the *regulator* of $K$ is
$$R_K = \abs{\det\big(N_j \log \abs{\sigma_j(u_i)}\big)_{1 \le i, j \le r}}.$$
This is the absolute value of the [[determinant|00KDZRW4C0]] of an $r \times r$ matrix.
The matrix keeps the first $r$ of the $r + 1$ columns $j = 1, \ldots, r_1 + r_2$.
If $r = 0$, the regulator is $R_K = 1$.

## Remarks

The value does not depend on the choices.
Each row of the full $r \times (r+1)$ matrix sums to $0$, because $\prod_j \abs{\sigma_j(u)}^{N_j} = \abs{N_{K/\Q}(u)} = 1$ for a unit $u$; so deleting another column changes only the sign of the determinant.
Another system of fundamental units differs by a matrix in $\mathrm{GL}_r(\Z)$, which has determinant $\pm 1$.
Here $\log$ is the [[natural logarithm|04DXAFN2ES]].

Example: for $K = \Q(\sqrt{2})$, $r = 1$ and $u_1 = 1 + \sqrt{2}$, so $R_K = \abs{\log(1 + \sqrt{2})} \approx 0.881$.

Non-example: in $\Q(\sqrt{2})$, the unit $3 + 2\sqrt{2} = (1 + \sqrt{2})^2$ is not fundamental, since it does not generate the units modulo $\pm 1$.
Using it in the formula gives $2 \log(1 + \sqrt{2})$, which is twice the regulator.

For $\Q$ and for imaginary quadratic fields, $r = 0$ and $R_K = 1$.