Theorems · Theorem · number theory
MulChar.sum_eq_zero_of_ne_one
∀ {R : Type u_1} [inst : CommMonoid R] [inst_1 : Fintype R] {R' : Type u_2} [inst_2 : CommRing R'] [IsDomain R']
{χ : MulChar R R'}, χ ≠ 1 → ∑ a, χ a = 0The sum over all values of a nontrivial multiplicative character on a finite ring is zero (when the target is a domain).
- Defined in
- Mathlib.NumberTheory.MulChar.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Fintypestatement and proof · cited by 7,736
- Finset.sumstatement and proof · cited by 5,195
- Finset.univstatement and proof · cited by 3,473
- Unitsproof · cited by 2,804
- Finset.sum_congrproof · cited by 2,323
- CommMonoidstatement and proof · cited by 2,264
- IsDomainstatement and proof · cited by 2,196
- Units.valproof · cited by 1,966
- Finset.mul_sumproof · cited by 196
- MulCharstatement and proof · cited by 186
Cited by9
Results whose statement or proof uses this declaration.
- DirichletCharacter.differentiableAt_LFunctionproof · cited by 4
- jacobiSum_mul_nontrivialproof · cited by 3
- DirichletCharacter.differentiableAt_completedLFunctionproof · cited by 1
- jacobiSum_one_nontrivialproof · cited by 1
- quadraticChar_sum_zeroproof · cited by 0
- jacobiSum_nontrivial_invproof · cited by 0
- DirichletCharacter.IsPrimitive.completedLFunction_one_subproof · cited by 0
- exists_jacobiSum_eq_neg_one_addproof · cited by 0
- gaussSum_one_rightproof · cited by 0