Dirichlet's unit theorem for orders

Proved

DirichletUnitTheorem.dirichlet_unit_theorem_order

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebraic-number-theorynumber-theory

Let $K$ be a number field and let $\mathcal O\subseteq\mathcal O_K$ be an order, i.e. a subring of $\mathcal O_K$ of finite index (as an additive subgroup). Then $\mathcal O^\times$ is finitely generated of rank

\operatorname{rank}\mathcal O^\times = r_1 + r_2 - 1.

So the unit theorem applies not only to the maximal order $\mathcal O_K$ but to every order.

Formalization Note An order is encoded as a subring of $\mathcal O_K$ whose additive group has finite index; every order of $K$ is of this form.

Preamble

import Mathlib
open NumberField

Formal statement

namespace DirichletUnitTheorem

theorem dirichlet_unit_theorem_order (K : Type*) [Field K] [NumberField K]
    (O : Subring (𝓞 K)) (hO : O.toAddSubgroup.FiniteIndex) :
    Group.FG Oˣ ∧
      Module.rank ℤ (Additive Oˣ) =
        ((InfinitePlace.nrRealPlaces K + InfinitePlace.nrComplexPlaces K - 1 : ℕ) : Cardinal) := by sorry

end DirichletUnitTheorem

Source

Wikipedia, "Dirichlet's unit theorem" (article supplied by the account owner as a PDF), lead section ("The theorem not only applies to the maximal order OK but to any order O ⊂ OK").

Read-back

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Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted this Lean statement, at the explicit request of the account owner. Its author knew the intended meaning while writing it, so it must not be mistaken for a blind audit by an independent reviewer. Please compare it against the Lean code itself.

Throughout, $K$ is a number field (a field of characteristic zero that is finite-dimensional over $\mathbb Q$ ), $\mathcal O_K$ is its ring of integers and $\mathcal O_K^\times$ the unit group. $r_1$ denotes the number of real infinite places of $K$ and $r_2$ the number of complex infinite places (each complex place corresponds to a pair of complex-conjugate embeddings $K\to\mathbb C$ ). The rank of an abelian group $A$ is its rank as a $\mathbb Z$ -module: the supremum of the cardinalities of $\mathbb Z$ -linearly independent families (for the multiplicative group $\mathcal O_K^\times$ : multiplicatively independent families). Torsion elements never belong to such a family. Statement. Let $K$ be a number field and $\mathcal O$ a subring of $\mathcal O_K$ whose underlying additive subgroup has finite index in $\mathcal O_K$ . Then the unit group $\mathcal O^\times$ of the ring $\mathcal O$ is finitely generated, and its rank (a cardinal) equals the natural number $r_1+r_2-1$ (truncated subtraction; $r_1+r_2\ge1$ always).

Human review

Endorsed by Shuze Chen · Oct 10, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 10, 2026
Confirmed by the mission captain (proposal self-audit).

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