Dirichlet's Unit Theorem

💡 The Core Insight

The unit group of a number field is finitely generated with rank determined by the number of real and complex embeddings, but always one less than the total dimension—a defect arising from the multiplicativity of the norm map.

Definition

Dirichlet's Unit Theorem: Let $K$ be a number field of degree $n = [K:\mathbb{Q}]$ with $r_1$ real embeddings and $r_2$ pairs of complex conjugate embeddings (so $n = r_1 + 2r_2$ ). Let $\mathcal{O}_K$ denote the ring of integers of $K$ and $\mathcal{O}_K^{\times}$ its unit group. Then there exists a finite group $\mu_K$ of roots of unity in $K$ and a free abelian group of rank $r = r_1 + r_2 - 1$ such that

\mathcal{O}_K^{\times} \cong \mu_K \times \mathbb{Z}^r

The integer $r$ is called the unit rank of $K$ , and a choice of generators $\varepsilon_1, \ldots, \varepsilon_r$ of the free part is called a system of fundamental units.

TheoremMain Result

Theorem (Dirichlet's Unit Theorem): Let $K$ be a number field with $r_1$ real embeddings $\sigma_1, \ldots, \sigma_{r_1}: K \to \mathbb{R}$ and $r_2$ pairs of complex embeddings $\tau_1, \overline{\tau_1}, \ldots, \tau_{r_2}, \overline{\tau_{r_2}}: K \to \mathbb{C}$ . Define the logarithmic embedding

L: K^{\times} \to \mathbb{R}^{r_1 + r_2}, \quad u \mapsto (\log|\sigma_1(u)|, \ldots, \log|\sigma_{r_1}(u)|, \log|\tau_1(u)|^2, \ldots, \log|\tau_{r_2}(u)|^2)

Then: [1] The image $L(\mathcal{O}_K^{\times})$ is a lattice in the hyperplane $H = \{(x_1, \ldots, x_{r_1+r_2}) : \sum x_i = 0\}$ of dimension $r_1 + r_2 - 1$ . [2] The kernel of $L$ restricted to $\mathcal{O}_K^{\times}$ is precisely $\mu_K$ , the group of roots of unity in $K$ . [3] Therefore $\mathcal{O}_K^{\times} \cong \mu_K \times \mathbb{Z}^{r_1 + r_2 - 1}$ .

Proof

We prove the theorem in several steps.

Step 1: The kernel of $L$ on units is finite. If $u \in \mathcal{O}_K^{\times}$ satisfies $L(u) = 0$ , then $|\sigma_i(u)| = 1$ for all $i$ and $|\tau_j(u)| = 1$ for all $j$ . Since $u$ and $u^{-1}$ are both algebraic integers, $u$ has all conjugates on the unit circle, so $u$ is a root of unity by Kronecker's theorem. Thus $\ker(L|_{\mathcal{O}_K^{\times}}) \subseteq \mu_K$ , which is finite since $K$ has only finitely many roots of unity (they form a finite cyclic group).

Step 2: The image lies in the hyperplane $H$ . For any $u \in K^{\times}$ , the norm $N_{K/\mathbb{Q}}(u) = \prod_{i=1}^{r_1} \sigma_i(u) \cdot \prod_{j=1}^{r_2} \tau_j(u)\overline{\tau_j(u)}$ . Taking absolute values and logarithms:

\log|N_{K/\mathbb{Q}}(u)| = \sum_{i=1}^{r_1} \log|\sigma_i(u)| + \sum_{j=1}^{r_2} \log|\tau_j(u)|^2

For $u \in \mathcal{O}_K^{\times}$ , we have $N_{K/\mathbb{Q}}(u) \in \mathbb{Z}^{\times} = \{\pm 1\}$ , so this sum equals 0. Therefore $L(\mathcal{O}_K^{\times}) \subseteq H$ .

Step 3: $L(\mathcal{O}_K^{\times})$ is discrete. Consider the preimage $L^{-1}(B)$ where $B$ is a bounded set in $\mathbb{R}^{r_1+r_2}$ . If $u \in \mathcal{O}_K^{\times}$ satisfies $L(u) \in B$ , then all conjugates of $u$ are bounded: $|\sigma_i(u)| \leq C$ and $|\tau_j(u)| \leq C$ for some constant $C$ . The coefficients of the minimal polynomial of $u$ are elementary symmetric functions of the conjugates, hence bounded. Since $u$ is an algebraic integer, there are only finitely many possibilities for $u$ (algebraic integers of bounded degree with bounded coefficients form a finite set). Thus $L^{-1}(B) \cap \mathcal{O}_K^{\times}$ is finite, proving discreteness.

Step 4: $L(\mathcal{O}_K^{\times})$ spans $H$ (the crucial step). We must show that $L(\mathcal{O}_K^{\times})$ is not contained in any proper subspace of $H$ . Suppose for contradiction that $L(\mathcal{O}_K^{\times})$ lies in a hyperplane $H' \subset H$ defined by $\sum_{i=1}^{r_1} a_i x_i + \sum_{j=1}^{r_2} b_j x_{r_1+j} = 0$ where not all coefficients are proportional to $(1, \ldots, 1)$ . We use Minkowski's theorem on successive minima. Consider the Minkowski space

M_K = \{(z_1, \ldots, z_{r_1}, w_1, \ldots, w_{r_2}) : z_i \in \mathbb{R}, w_j \in \mathbb{C}\}

with embedding $\sigma: K \to M_K$ given by $\alpha \mapsto (\sigma_1(\alpha), \ldots, \sigma_{r_1}(\alpha), \tau_1(\alpha), \ldots, \tau_{r_2}(\alpha))$ . For any tuple of positive reals $(t_1, \ldots, t_{r_1}, s_1, \ldots, s_{r_2})$ with $\prod t_i \prod s_j^2 > (\frac{n!}{n^n})^{1/2} \sqrt{|d_K|}$ where $d_K$ is the discriminant, Minkowski's theorem guarantees a nonzero $\alpha \in \mathcal{O}_K$ with $|\sigma_i(\alpha)| < t_i$ and $|\tau_j(\alpha)| < s_j$ . By choosing the $t_i$ and $s_j$ appropriately (with ratios corresponding to desired ratios in the logarithmic image), we can construct infinitely many units whose logarithmic images point in directions violating the supposed restriction to $H'$ . This contradiction shows $L(\mathcal{O}_K^{\times})$ spans $H$ .

Step 5: Conclusion. A discrete subgroup of $\mathbb{R}^m$ that spans $\mathbb{R}^m$ is a lattice of rank $m$ . Since $L(\mathcal{O}_K^{\times})$ is discrete in $H \cong \mathbb{R}^{r_1+r_2-1}$ and spans it, $L(\mathcal{O}_K^{\times})$ is a lattice of rank $r_1 + r_2 - 1$ . By the structure theorem for finitely generated abelian groups and Step 1, we have $\mathcal{O}_K^{\times} \cong \mu_K \times \mathbb{Z}^{r_1+r_2-1}$ . □

∎

⚠️ Common mistake

A common error is to expect the unit rank to be $r_1 + r_2$ (the total number of embeddings up to conjugacy) rather than $r_1 + r_2 - 1$ . The "missing dimension" comes from the constraint that the norm of a unit must be $\pm 1$ , which imposes one independent multiplicative relation among the absolute values of conjugates. This is not a defect of the proof but a genuine arithmetic phenomenon!

mathematician

Peter Gustav Lejeune Dirichlet (1846) — Proved the unit theorem using his analytic class number formula, establishing the fundamental connection between the residue of the Dedekind zeta function and the unit regulator.

visualization

Example1

Computing units in $\mathbb{Q}(\sqrt{2})$ For $K = \mathbb{Q}(\sqrt{2})$ , we have $r_1 = 2$ (embeddings $\sigma_1: \sqrt{2} \mapsto \sqrt{2}$ and $\sigma_2: \sqrt{2} \mapsto -\sqrt{2}$ ) and $r_2 = 0$ . Thus the unit rank is $r = 2 + 0 - 1 = 1$ . The roots of unity in $\mathbb{Q}(\sqrt{2})$ are just $\mu_K = \{\pm 1\}$ since any root of unity in a real field must be $\pm 1$ . We seek a fundamental unit. Consider $\varepsilon = 1 + \sqrt{2}$ . We verify it's a unit: $(1+\sqrt{2})(1-\sqrt{2}) = 1 - 2 = -1$ , so $\varepsilon^{-1} = -(1-\sqrt{2}) = -1 + \sqrt{2} \in \mathcal{O}_K$ . To confirm $\varepsilon$ is fundamental, we check that no proper power of it equals $\pm 1$ . Computing: $\varepsilon^2 = 3 + 2\sqrt{2}$ , which is not $\pm 1$ . By Pell's equation theory, $1 + \sqrt{2}$ is indeed the fundamental unit. Therefore $\mathcal{O}_K^{\times} = \{\pm(1+\sqrt{2})^n : n \in \mathbb{Z}\} \cong \{\pm 1\} \times \mathbb{Z}$ .

Example2

Units in the <span class="wiki-link-missing">Eisenstein integers</span> $\mathbb{Z}[\omega]$ where $\omega = e^{2\pi i/3}$ Let $K = \mathbb{Q}(\omega)$ where $\omega$ is a primitive cube root of unity. Then $[K:\mathbb{Q}] = 2$ , with $r_1 = 0$ and $r_2 = 1$ (one pair of complex conjugate embeddings). The unit rank is $r = 0 + 1 - 1 = 0$ . The theorem predicts $\mathcal{O}_K^{\times} \cong \mu_K \times \mathbb{Z}^0 = \mu_K$ is finite. Indeed, the roots of unity in $K$ are $\mu_K = \{1, \omega, \omega^2, -1, -\omega, -\omega^2\}$ , a cyclic group of order 6. We verify these are the only units. If $u = a + b\omega \in \mathcal{O}_K^{\times}$ with $a,b \in \mathbb{Z}$ , then $N(u) = u \cdot \bar{u} = (a+b\omega)(a+b\omega^2) = a^2 - ab + b^2 = \pm 1$ . The integer solutions are $(a,b) \in \{(\pm 1, 0), (0, \pm 1), (\pm 1, \pm 1)\}$ , giving exactly the six units listed. This illustrates that when there are no real embeddings and the unit rank vanishes, the unit group is purely torsion.

Counterexample

Why $\mathcal{O}_K^{\times}$ need not be finitely generated without the number field assumption Consider $R = \mathbb{Z}[x]$ , the polynomial ring. This is an integral domain but not the ring of integers of a number field. The unit group is $R^{\times} = \{\pm 1\}$ , which is finitely generated. However, consider $R = \mathbb{Z}[x, 1/x] = \mathbb{Z}[x^{\pm 1}]$ , the Laurent polynomial ring. Here $R^{\times} = \{\pm x^n : n \in \mathbb{Z}\} \cong \{\pm 1\} \times \mathbb{Z}$ , still finitely generated. But now consider the ring $R = k[x_1, x_2, x_3, \ldots]$ where $k$ is a field—an infinite polynomial ring. The unit group is just $R^{\times} = k^{\times}$ , which may or may not be finitely generated depending on $k$ . More dramatically, the coordinate ring of an algebraic variety with infinitely many components can have units forming an infinitely generated group. The finiteness in Dirichlet's theorem crucially uses that $K$ is a finite extension of $\mathbb{Q}$ and that $\mathcal{O}_K$ is a Dedekind domain with finitely many primes above any rational prime.

🧠 Intuition

Geometric picture: Think of units as "volume-preserving transformations" in number space. The logarithmic embedding $L$ converts multiplicative structure to additive structure, turning units into lattice points. The norm condition $|N(u)| = 1$ becomes the constraint that the product of all coordinate-stretches equals 1—a volume-preservation condition. This forces units to lie on a hyperplane of codimension 1. In $\mathbb{Q}(\sqrt{2})$ , a unit $u = a + b\sqrt{2}$ acts on the two "directions" (the two real embeddings). The fundamental unit $1 + \sqrt{2} \approx 2.414$ expands by factor ~2.414 in one direction and contracts by factor ~0.414 in the other direction, preserving the product. All units are powers of this basic "hyperbolic rotation" together with sign changes. In cyclotomic fields and other fields with complex embeddings, units can perform genuine rotations (coming from roots of unity) combined with rescalings in the real hyperbolic part of the hyperplane.

connection

Arakelov geometry and heights: The logarithmic embedding in Dirichlet's theorem is the foundation for Arakelov theory, which completes the set of places of a number field by adding archimedean places. The regulator can be interpreted as a volume in Arakelov-Chow group theory. Moreover, the logarithmic embedding is essential in defining height functions on algebraic varieties—the Néron-Tate height on abelian varieties uses the fact that rational points modulo torsion form a finitely generated group (the Mordell-Weil theorem), directly analogous to Dirichlet's unit theorem. Both theorems reflect fundamental finiteness phenomena in arithmetic geometry stemming from the product formula.

Dirichlet's Unit Theorem

Overview

Core Theory

Examples and Applications

Historical Development

Connections to Other Areas

What You Need First

What This Opens Up

See Also