Theorems · Theorem · number theory
gaussSum_ne_zero_of_nontrivial
∀ {R : Type u} [inst : Field R] [inst_1 : Fintype R] {R' : Type v} [inst_2 : CommRing R'] [IsDomain R'],
↑(Fintype.card R) ≠ 0 → ∀ {χ : MulChar R R'}, χ ≠ 1 → ∀ {ψ : AddChar R R'}, ψ.IsPrimitive → gaussSum χ ψ ≠ 0The Gauss sum of a nontrivial character on a finite field does not vanish.
- Defined in
- Mathlib.NumberTheory.GaussSum
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Fintypestatement and proof · cited by 7,736
- Fieldstatement and proof · cited by 7,404
- IsDomainstatement and proof · cited by 2,196
- MulZeroClass.zero_mulproof · cited by 1,625
- Fintype.cardstatement and proof · cited by 1,386
- AddCharstatement and proof · cited by 286
- MulCharstatement and proof · cited by 186
- gaussSumstatement and proof · cited by 29
- AddChar.IsPrimitivestatement and proof · cited by 22
- gaussSum_mul_gaussSum_eq_cardproof · cited by 4
Cited by1
Results whose statement or proof uses this declaration.
- jacobiSum_eq_gaussSum_mul_gaussSum_div_gaussSumproof · cited by 0