Dirichlet's Unit Theorem - Core Definitions

Dirichlet's Unit Theorem describes the structure of the unit group in rings of integers, one of the deepest results in classical algebraic number theory.

DefinitionUnits in Number Fields

An element $\epsilon \in \mathcal{O}_K$ is a unit if it has a multiplicative inverse in $\mathcal{O}_K$ . Equivalently, $\epsilon$ is a unit if and only if $N_{K/\mathbb{Q}}(\epsilon) = \pm 1$ .

The set of units $\mathcal{O}_K^*$ forms a group under multiplication. Understanding the structure of this group is fundamental to the arithmetic of $K$ .

ExampleUnits in Familiar Fields

Rationals: $\mathbb{Z}^* = \{\pm 1\}$ (finite, order 2)
Gaussian integers: $\mathbb{Z}[i]^* = \{\pm 1, \pm i\}$ (finite, order 4)
Real quadratic fields: $\mathbb{Z}[\sqrt{2}]^* = \{\pm (1+\sqrt{2})^n : n \in \mathbb{Z}\}$ (infinite!)

The transition from finite to infinite unit groups occurs precisely when $K$ has real embeddings.

DefinitionRoots of Unity

The roots of unity in $K$ are the elements $\zeta \in \mathcal{O}_K$ such that $\zeta^n = 1$ for some $n \geq 1$ . These form a finite cyclic group, denoted $\mu_K$ .

For most number fields, $\mu_K = \{\pm 1\}$ . Exceptions include:

$\mu_{\mathbb{Q}(i)} = \{\pm 1, \pm i\}$ (order 4)
$\mu_{\mathbb{Q}(\sqrt{-3})} = \{\pm 1, \pm \omega, \pm \omega^2\}$ where $\omega = e^{2\pi i/3}$ (order 6)
$\mu_{\mathbb{Q}(\zeta_n)} = \langle \zeta_n \rangle$ (order $n$ )

TheoremDirichlet's Unit Theorem (Statement)

Let $K$ be a number field with $r_1$ real embeddings and $r_2$ pairs of complex conjugate embeddings. Then: $\mathcal{O}_K^* \cong \mu_K \times \mathbb{Z}^{r_1 + r_2 - 1}$

where $\mu_K$ is the finite cyclic group of roots of unity in $K$ , and the free part has rank $r = r_1 + r_2 - 1$ .

Equivalently, there exist fundamental units $\epsilon_1, \ldots, \epsilon_r$ such that every unit $\epsilon \in \mathcal{O}_K^*$ can be uniquely written as: $\epsilon = \zeta \epsilon_1^{a_1} \cdots \epsilon_r^{a_r}$ where $\zeta \in \mu_K$ and $a_i \in \mathbb{Z}$ .

Remark

The unit rank $r = r_1 + r_2 - 1$ depends only on the signature of $K$ :

Imaginary quadratic fields ( $r_1 = 0, r_2 = 1$ ): $r = 0$ , so $\mathcal{O}_K^* = \mu_K$ is finite
Real quadratic fields ( $r_1 = 2, r_2 = 0$ ): $r = 1$ , one fundamental unit
Cubic totally real ( $r_1 = 3, r_2 = 0$ ): $r = 2$ , two fundamental units
Cyclotomic $\mathbb{Q}(\zeta_p)$ ( $r_1 = 0, r_2 = (p-1)/2$ ): $r = (p-3)/2$

DefinitionFundamental Unit

For a real quadratic field $K = \mathbb{Q}(\sqrt{d})$ with $d > 0$ , the fundamental unit $\epsilon_0$ is the smallest unit $> 1$ . All other units are of the form $\pm \epsilon_0^n$ for $n \in \mathbb{Z}$ .

Computing $\epsilon_0$ is equivalent to solving Pell's equation $x^2 - dy^2 = \pm 1$ : if $\epsilon_0 = x_0 + y_0\sqrt{d}$ , then $(x_0, y_0)$ is the fundamental solution.