Squared modulus of a Gauss sum

Proved

Weil.norm_gaussSum_sq

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

exponential-sumsfinite-fieldsgauss-sumsnumber-theoryweil-bound

For a finite field $F$ , a non-trivial multiplicative character $\chi : F \to \mathbb{C}$ and a primitive additive character $\psi$ , the Gauss sum $g(\chi,\psi)=\sum_{a\in F}\chi(a)\psi(a)$ satisfies

\lVert g(\chi,\psi)\rVert^2 = \#F.

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_gaussSum_sq : ∀ {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‖ ^ 2 = (↑(Fintype.card F) : ℝ) := 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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