$n = r_1 + 2r_2$

Proved

DirichletUnitTheorem.degree_eq_nrRealPlaces_add_two_mul_nrComplexPlaces

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

algebraic-number-theorynumber-theory

Let $K$ be a number field of degree $n=[K:\mathbb Q]$ , with $r_1$ real embeddings and $r_2$ conjugate pairs of non-real complex embeddings. Then

n = r_1 + 2r_2.

This is the counting identity behind the definition of $r_1$ and $r_2$ : the $n$ embeddings $K\to\mathbb C$ are either real or come in complex-conjugate pairs.

Preamble

import Mathlib
open NumberField

Formal statement

namespace DirichletUnitTheorem

theorem degree_eq_nrRealPlaces_add_two_mul_nrComplexPlaces (K : Type*) [Field K]
    [NumberField K] :
    InfinitePlace.nrRealPlaces K + 2 * InfinitePlace.nrComplexPlaces K = Module.finrank ℚ K := by sorry

end DirichletUnitTheorem

Source

Wikipedia, "Dirichlet's unit theorem" (article supplied by the account owner as a PDF), lead section, identity n = r1 + 2r2.

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

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 $K\to\mathbb C$ ). Statement. For every number field $K$ ,

r_1 + 2r_2 = [K:\mathbb Q],

where $[K:\mathbb Q]$ is the dimension of $K$ as a $\mathbb Q$ -vector space. No other hypotheses.

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).

View graph

n=r1+2r2n = r_1 + 2r_2n=r1​+2r2​

$n = r_1 + 2r_2$