Units and Dirichlet Unit Theorem

Multiplicative Invertibility

Let $K$ be a number field and let

\mathcal O_K

be its ring of integers.

An element

u\in\mathcal O_K

is called a unit if it has a multiplicative inverse inside $\mathcal O_K$ . That is, there exists

v\in\mathcal O_K

such that

uv=1.

The set of all units forms a multiplicative group denoted

\mathcal O_K^\times.

Units generalize the familiar invertible integers

\pm1

inside $\mathbb{Z}$ .

Understanding units is one of the main structural problems in algebraic number theory.

Units in Familiar Rings

Ordinary Integers

\mathbb{Z},

the only units are

1 \quad\text{and}\quad -1.

Indeed, if

ab=1

for integers $a,b$ , then both must have absolute value $1$ .

Gaussian Integers

\mathbb{Z}[i],

the units are

\pm1, \qquad \pm i.

Their norms are

1.

No other Gaussian integer has norm $1$ .

Real Quadratic Fields

In real quadratic fields, infinitely many units may exist.

For example, in

\mathbb{Q}(\sqrt2),

the element

1+\sqrt2

is a unit because

(1+\sqrt2)(-1+\sqrt2)=1.

Its powers produce infinitely many distinct units.

Thus unit groups in number fields can be dramatically richer than in $\mathbb{Z}$ .

Norm Criterion

The norm detects units.

u\in\mathcal O_K,

then $u$ is a unit if and only if

N_{K/\mathbb{Q}}(u)=\pm1.

Indeed, if

uv=1,

then multiplicativity gives

N(u)N(v)=1.

Since norms of algebraic integers are ordinary integers, both norms must equal $\pm1$ .

Conversely, if

N(u)=\pm1,

then the inverse of $u$ can be expressed using its conjugates and remains an algebraic integer.

Thus units are precisely the norm- $\pm1$ elements.

Pell Equations and Units

Pell equations are directly connected with units in quadratic fields.

Consider

x^2-Dy^2=1.

u=x+y\sqrt D,

then

N(u)=x^2-Dy^2.

Thus solving the Pell equation amounts to finding units of norm $1$ in

\mathbb{Q}(\sqrt D).

For example,

3+2\sqrt2

has norm

9-8=1.

Its powers generate infinitely many Pell equation solutions.

Hence continued fractions, Pell equations, and unit groups are closely related.

Logarithmic Embedding

The structure of the unit group becomes clearer through logarithms.

Suppose $K$ has:

$r_1$ real embeddings,
$r_2$ pairs of complex embeddings.

Define the logarithmic map

u\mapsto ( \log|\sigma_1(u)|, \dots, \log|\sigma_{r_1+r_2}(u)| ).

Because norms of units equal $\pm1$ , these logarithms satisfy one linear relation:

\sum \log|\sigma_i(u)|=0.

Thus the logarithmic images lie inside a hyperplane.

This converts multiplicative arithmetic into additive geometry.

Dirichlet Unit Theorem

The fundamental structure theorem is the following.

Theorem (Dirichlet Unit Theorem). Let $K$ be a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings. Then

\mathcal O_K^\times \cong \mu(K)\times\mathbb{Z}^{\,r_1+r_2-1},

where:

$\mu(K)$ is the finite group of roots of unity in $K$ ,
the free abelian part has rank

r_1+r_2-1.



\mathcal O_K^\times\cong \mu(K)\times \mathbb Z^{r_1+r_2-1}

This theorem completely describes the structure of the unit group.

Consequences

Imaginary Quadratic Fields

K=\mathbb{Q}(\sqrt{-d}),

then

r_1=0, \qquad r_2=1.

Hence the rank is

0+1-1=0.

Thus the unit group is finite.

This explains why rings like

\mathbb{Z}[i]

have only finitely many units.

Real Quadratic Fields

K=\mathbb{Q}(\sqrt d), \qquad d>0,

then

r_1=2, \qquad r_2=0.

Hence the rank is

2-1=1.

Therefore the unit group is infinite cyclic up to sign:

\mathcal O_K^\times \cong \{\pm1\}\times\mathbb Z.

This explains the infinite families of Pell equation solutions.

Fundamental Units

In real quadratic fields, the free part of the unit group is generated by a single unit

\varepsilon>1,

called the fundamental unit.

Every unit has the form

\pm \varepsilon^n.

For example, in

\mathbb{Q}(\sqrt2),

one may take

\varepsilon=1+\sqrt2.

Then every unit is

\pm(1+\sqrt2)^n.

Computing fundamental units is a major computational problem in algebraic number theory.

Regulator

The logarithmic embedding produces a lattice.

The covolume of this lattice is called the regulator of the field.

The regulator measures the arithmetic size of the unit group.

Large regulators correspond to units that are spread far apart geometrically.

The regulator appears in analytic formulas involving zeta functions and class numbers.

Arithmetic Importance

Units play a central role in:

Diophantine equations,
factorization,
ideal arithmetic,
Pell equations,
class number formulas.

In many arguments, units represent the main source of arithmetic flexibility.

For example, solutions of norm equations often differ only by multiplication by units.

Thus understanding units is essential for understanding arithmetic structure.

Geometric Interpretation

Under the logarithmic embedding, units become lattice points in Euclidean space.

Dirichlet theorem therefore states that the unit group has geometric lattice structure.

This interpretation links algebraic number theory with geometry of numbers and Lie group methods.

Arithmetic multiplicative structure becomes linear geometry after taking logarithms.

Historical Perspective

The Dirichlet unit theorem was proved by entity[“people”,“Peter Gustav Lejeune Dirichlet”,“German mathematician”] and became one of the foundational results of algebraic number theory.

It revealed that unit groups possess rigid algebraic structure controlled entirely by the embeddings of the number field.

The theorem is one of the earliest major examples where algebra, analysis, and geometry interact deeply inside arithmetic.