Theorems · Theorem · number theory

factorsThrough_of_gaussSum_ne_zero

∀ {N : ℕ} [inst : NeZero N] {R : Type u_1} [inst_1 : CommRing R] (e : AddChar (ZMod N) R) [IsDomain R]
  {χ : DirichletCharacter R N} {d : ℕ}, d ∣ N → e.mulShift ↑d = 1 → gaussSum χ e ≠ 0 → χ.FactorsThrough d

If gaussSum χ e ≠ 0, and d is such that e.mulShift d = 1, then χ must factor through d. (This will be used to show that Gauss sums vanish when χ is primitive and e is not.)

Defined in

Mathlib.NumberTheory.DirichletCharacter.GaussSum

Cited by

1 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NeZeroCommRingIsDomain