A character of order $\gcd(\#F^\times, d)$ has trivial $\gcd$ -th power

Proved

Weil.pow_gcdDeg_eq_one

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

exponential-sumsfinite-fieldsgauss-sumsnumber-theoryweil-bound

Let $F$ be a finite field, $d \in \mathbb{N}$ , and $\chi$ a multiplicative character of $F$ valued in $\mathbb{C}$ whose order is $m = \gcd(\#F^\times, d)$ . Then $\chi^m = 1$ .

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 pow_gcdDeg_eq_one : ∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F] {d : ℕ} {χ : MulChar F ℂ}, orderOf χ = Nat.gcd (Fintype.card Fˣ) d → χ ^ Nat.gcd (Fintype.card Fˣ) d = 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)

View graph

A character of order gcd⁡(#F×,d)\gcd(\#F^\times, d)gcd(#F×,d) has trivial gcd⁡\gcdgcd-th power

A character of order $\gcd(\#F^\times, d)$ has trivial $\gcd$ -th power