$\sum_w N_w\log\lvert u\rvert_w = 0$ for every unit

Proved

DirichletUnitTheorem.sum_mult_log_eq_zero

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

algebraic-number-theorynumber-theory

Let $K$ be a number field and $u\in\mathcal O_K^\times$ . Writing $\sigma_j$ for the embeddings corresponding to the infinite places and $N_j\in\{1,2\}$ for the local degrees,

\sum_{j} N_j\log\lvert\sigma_j(u)\rvert = 0.

The left side is $\log\lvert N_{K/\mathbb Q}(u)\rvert$ , and units have norm $\pm1$ . This is the reason the rows of the regulator matrix sum to zero.

Preamble

import Mathlib
open NumberField

Formal statement

namespace DirichletUnitTheorem

theorem sum_mult_log_eq_zero (K : Type*) [Field K] [NumberField K] (u : (𝓞 K)ˣ) :
    ∑ w : InfinitePlace K, (w.mult : ℝ) * Real.log (w ((u : 𝓞 K) : K)) = 0 := by sorry

end DirichletUnitTheorem

Source

Wikipedia, "Dirichlet's unit theorem" (article supplied by the account owner as a PDF), section "The regulator" ("the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries in a row)").

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 $K\to\mathbb C$ ). For an infinite place $w$ of $K$ , $\lvert x\rvert_w$ denotes the absolute value $\lvert\sigma(x)\rvert$ for an embedding $\sigma$ inducing $w$ , and $N_w$ is $1$ if $w$ is real and $2$ if $w$ is complex. $\log$ is the real logarithm (Mathlib's convention gives $\log 0 = 0$ , but units have $\lvert u\rvert_w>0$ ). Statement. For every number field $K$ and every unit $u\in\mathcal O_K^\times$ ,

\sum_{w} N_w \log\lvert u\rvert_w = 0,

the sum running over all infinite places $w$ of $K$ .

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

∑wNwlog⁡∣u∣w=0\sum_w N_w\log\lvert u\rvert_w = 0∑w​Nw​log∣u∣w​=0 for every unit

$\sum_w N_w\log\lvert u\rvert_w = 0$ for every unit