Theorems · Theorem · number theory
gaussSum_pow_eq_prod_jacobiSum
∀ {F : Type u_1} {R : Type u_2} [inst : Fintype F] [inst_1 : Field F] [inst_2 : CommRing R] [IsDomain R]
{χ : MulChar F R} {ψ : AddChar F R},
2 ≤ orderOf χ →
ψ.IsPrimitive →
gaussSum χ ψ ^ orderOf χ = χ (-1) * ↑(Fintype.card F) * ∏ i ∈ Finset.Ico 1 (orderOf χ - 1), jacobiSum χ (χ ^ i)If χ is a multiplicative character of order n ≥ 2 on a finite field F,
then g(χ)^n = χ(-1) * #F * J(χ,χ) * J(χ,χ²) * ... * J(χ,χⁿ⁻²).
- Defined in
- Mathlib.NumberTheory.JacobiSum.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites21
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- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Fintypestatement and proof · cited by 7,736
- Fieldstatement and proof · cited by 7,404
- Finset.prodstatement and proof · cited by 2,356
- IsDomainstatement and proof · cited by 2,196
- mul_assocproof · cited by 1,667
- Fintype.cardstatement and proof · cited by 1,386
- LE.le.trans_ltproof · cited by 795
- Finset.Icostatement and proof · cited by 450
- orderOfstatement and proof · cited by 324
- AddCharstatement and proof · cited by 286
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