Theorem

Dirichlet's Unit Theorem

Number Theory

Dirichlet's Unit Theorem describes the structure of the group of units, meaning the invertible elements, in the ring of integers of an algebraic number field, showing that the group is a direct product of a finite cyclic group of roots of unity and a free abelian group whose rank is determined by the number of real and complex embeddings of the field. Named for Peter Gustav Lejeune Dirichlet, it is a foundational result of algebraic number theory.

Facts

Statement

The group of units in the ring of integers of a number field K is finitely generated, with rank equal to r1 plus r2 minus 1, where r1 is the number of real embeddings of K and r2 is the number of conjugate pairs of complex embeddings of K. 1

Classification

Statement Form

Identity or Equation 1

Connections

Has Statement Form

Equation

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Identity

In Branch

Number Theory

Proved By

Peter Gustav Lejeune Dirichlet

Source Dirichlet's Unit Theorem (Wikipedia)

Sources

1. Dirichlet's Unit Theorem (Wikipedia)

Wikimedia Foundation

opening paragraph
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K.
Proved By: Peter Gustav Lejeune Dirichlet, Lead paragraph
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank

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Dirichlet's Unit Theorem

Has Statement Form

In Branch

Proved By