Multiplicative characters are unimodular at units

Proved

Weil.norm_mulChar_apply

by tabbott · Sep 2, 2026 · Mathlib c5ea003 (Lean v4.30.0)

exponential-sumsfinite-fieldsgauss-sumsnumber-theoryweil-bound

Let $F$ be a finite field and $\chi$ a multiplicative character of $F$ valued in $\mathbb{C}$ . For $u \in F$ with $u \ne 0$ , $\lVert\chi(u)\rVert = 1$ , because $\chi(u)^{\#F^\times} = 1$ makes $\chi(u)$ a root of unity.

Preamble

import Mathlib.NumberTheory.GaussSum
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
import Mathlib.NumberTheory.MulChar.Lemmas
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.GroupTheory.Index
import Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Algebra.Field.GeomSum
import Mathlib.Algebra.Order.BigOperators.Group.Finset

set_option autoImplicit false
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false

universe u_1 u_2

open AddChar MulChar Finset

Formal statement

namespace Weil
theorem norm_mulChar_apply : ∀ {F : Type u_1} [inst : Field F] [Fintype F] [DecidableEq F] (χ : MulChar F ℂ) {u : F}, u ≠ 0 → ‖(χ : F → ℂ) u‖ = 1 := by sorry
end Weil

Source

Ireland & Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer GTM 84, Ch. 8 (Gauss and Jacobi Sums)

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