Gauss sum times its conjugate equals the field size

Proved

Weil.gaussSum_mul_star

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

exponential-sumsfinite-fieldsgauss-sumsnumber-theoryweil-bound

Let $F$ be a finite field, $\chi$ a non-trivial multiplicative character of $F$ with values in $\mathbb{C}$ , and $\psi$ a primitive additive character of $F$ . Write $g(\chi,\psi)=\sum_{a\in F}\chi(a)\psi(a)$ for the associated Gauss sum. Then

g(\chi,\psi)\,\overline{g(\chi,\psi)} = \#F.

This is the algebraic core of the classical evaluation $|g(\chi,\psi)|=\sqrt{\#F}$ : it combines the identity $g(\chi,\psi)g(\chi^{-1},\psi^{-1})=\#F$ with the fact that complex conjugation inverts both characters.

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 gaussSum_mul_star : ∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] (χ : MulChar F ℂ), χ ≠ 1 → ∀ (ψ : AddChar F ℂ), AddChar.IsPrimitive ψ → gaussSum χ ψ * star (gaussSum χ ψ) = (↑(Fintype.card F) : ℂ) := 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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