Dirichlet's Unit Theorem - Key Properties

The logarithmic embedding and regulator provide geometric tools for understanding the unit group structure.

DefinitionLogarithmic Embedding

Let $\sigma_1, \ldots, \sigma_{r_1}$ be the real embeddings and $\tau_1, \bar{\tau}_1, \ldots, \tau_{r_2}, \bar{\tau}_{r_2}$ be the complex embeddings of $K$ . The logarithmic embedding is: $\lambda: \mathcal{O}_K^* \to \mathbb{R}^{r_1 + r_2}$ $\lambda(\epsilon) = (\log|\sigma_1(\epsilon)|, \ldots, \log|\sigma_{r_1}(\epsilon)|, 2\log|\tau_1(\epsilon)|, \ldots, 2\log|\tau_{r_2}(\epsilon)|)$

The factor of 2 for complex embeddings accounts for conjugate pairs. This map takes products to sums: $\lambda(\epsilon\eta) = \lambda(\epsilon) + \lambda(\eta)$ .

The image of $\lambda$ lies in the trace-zero hyperplane: $H = \left\{(x_1, \ldots, x_{r_1+r_2}) \in \mathbb{R}^{r_1+r_2} : \sum_{i=1}^{r_1} x_i + \sum_{j=1}^{r_2} x_j = 0\right\}$

This follows from $N_{K/\mathbb{Q}}(\epsilon) = \pm 1$ , giving $\sum_i \log|\sigma_i(\epsilon)| + 2\sum_j \log|\tau_j(\epsilon)| = 0$ .

DefinitionRegulator

Let $\epsilon_1, \ldots, \epsilon_r$ be fundamental units. The regulator $R_K$ is the absolute value of the determinant of any $(r \times r)$ minor of the $(r \times (r+1))$ matrix with rows $\lambda(\epsilon_i)$ .

Explicitly, for a real quadratic field with fundamental unit $\epsilon_0$ : $R_K = |\log|\epsilon_0||$

For general fields, $R_K$ measures the "volume" of the fundamental domain of the unit lattice in the trace-zero hyperplane.

ExampleComputing Regulators

For $K = \mathbb{Q}(\sqrt{2})$ , the fundamental unit is $\epsilon_0 = 1 + \sqrt{2}$ with norm $-1$ .

The two real embeddings are $\sigma_1(\epsilon_0) = 1 + \sqrt{2} \approx 2.414$ and $\sigma_2(\epsilon_0) = 1 - \sqrt{2} \approx -0.414$ .

The regulator is: $R_K = |\log|1 + \sqrt{2}|| = \log(1 + \sqrt{2}) \approx 0.881$

For $K = \mathbb{Q}(\sqrt{5})$ , fundamental unit $\epsilon_0 = \frac{1+\sqrt{5}}{2}$ (golden ratio): $R_K = \log\left(\frac{1+\sqrt{5}}{2}\right) \approx 0.481$

TheoremUnit Lattice

The image $\lambda(\mathcal{O}_K^*)$ is a lattice in the hyperplane $H \cong \mathbb{R}^r$ of rank $r = r_1 + r_2 - 1$ .

The fundamental units generate this lattice: $\lambda(\mathcal{O}_K^*) = \mathbb{Z}\lambda(\epsilon_1) + \cdots + \mathbb{Z}\lambda(\epsilon_r)$ .

The regulator $R_K$ equals the covolume of this lattice in $H$ .

Remark

The regulator appears in the class number formula: $h_K = \frac{w_K \sqrt{|\Delta_K|}}{2^{r_1}(2\pi)^{r_2} R_K} \lim_{s \to 1^+} (s-1)\zeta_K(s)$

Large regulators correspond to "sparse" unit groups (fundamental units with large norm), while small regulators indicate "dense" units. The product $h_K R_K$ appears in the Brauer-Siegel theorem.

ExampleCyclotomic Units

For $K = \mathbb{Q}(\zeta_p)$ with $p$ prime, explicit units are: $\epsilon_k = \frac{1 - \zeta_p^k}{1 - \zeta_p}, \quad k = 2, \ldots, \frac{p-1}{2}$

These cyclotomic units have rank $(p-3)/2$ , exactly the unit rank. They generate a subgroup of finite index in $\mathcal{O}_K^*$ , and the index (the unit index) divides the class number.