Intermediate powers of a character of known order are non-trivial

Proved

Weil.pow_ne_one_of_mem_Ico

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 $\mathbb{C}$ of order exactly $m = \gcd(\#F^\times, d)$ . Then $\chi^j \ne 1$ for every $j$ with $1 \le j < m$ .

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_ne_one_of_mem_Ico : ∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F] {d : ℕ} {χ : MulChar F ℂ} {j : ℕ}, j ∈ Finset.Ico 1 (Nat.gcd (Fintype.card Fˣ) d) → orderOf χ = Nat.gcd (Fintype.card Fˣ) d → χ ^ j ≠ 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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