Theorems · Theorem · complex analysis
Complex.mem_rootsOfUnity
∀ (n : ℕ) [NeZero n] (x : ℂˣ), x ∈ rootsOfUnity n ℂ ↔ ∃ i < n, Complex.exp (2 * ↑Real.pi * Complex.I * (↑i / ↑n)) = ↑x
The complex n-th roots of unity are exactly the
complex numbers of the form exp (2 * Real.pi * Complex.I * (i / n)) for some i < n.
- Defined in
- Mathlib.RingTheory.RootsOfUnity.Complex
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NeZero
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Complexstatement and proof · cited by 5,565
- mul_oneproof · cited by 3,885
- Subgroupstatement · cited by 3,593
- one_mulproof · cited by 2,841
- Unitsstatement and proof · cited by 2,804
- Units.valstatement and proof · cited by 1,966
- Nat.cast_zeroproof · cited by 1,870
- Real.pistatement and proof · cited by 1,774
- Complex.ofRealstatement and proof · cited by 1,654
- one_ne_zeroproof · cited by 885
- Complex.Istatement and proof · cited by 866
- div_oneproof · cited by 629
Cited by1
Results whose statement or proof uses this declaration.
- surjective_rootsOfUnityCircleEquiv_comp_rootsOfUnityAddCharproof · cited by 1