The complex-valued quadratic character is non-trivial

Proved

Weil.cchar_ne_one

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

exponential-sumsfinite-fieldsgauss-sumsnumber-theoryweil-bound

Let $F$ be a finite field whose characteristic is not $2$ . The quadratic (Legendre) character of $F$ , composed with the inclusion $\mathbb{Z} \hookrightarrow \mathbb{C}$ , is a non-trivial multiplicative character $F \to \mathbb{C}$ . Non-triviality is exactly the statement that not every non-zero element of $F$ is a square, which fails only in characteristic two.

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 cchar_ne_one : ∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F], ringChar F ≠ 2 → MulChar.ringHomComp (quadraticChar F) (Int.castRingHom ℂ) ≠ 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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