Complex roots of unity #

In this file we show that the n-th complex roots of unity are exactly the complex numbers exp (2 * π * I * (i / n)) for i ∈ Finset.range n.

Main declarations #

Complex.mem_rootsOfUnity: the complex n-th roots of unity are exactly the complex numbers of the form exp (2 * π * I * (i / n)) for some i < n.
Complex.card_rootsOfUnity: the number of n-th roots of unity is exactly n.
Complex.norm_rootOfUnity_eq_one: A complex root of unity has norm 1.

theorem Complex.isPrimitiveRoot_exp_of_coprime (i n : ℕ) (h0 : n ≠ 0) (hi : i.Coprime n) :

theorem Complex.isPrimitiveRoot_iff (ζ : ℂ) (n : ℕ) (hn : n ≠ 0) :

IsPrimitiveRoot ζ n ↔ ∃ i < n, ∃ (_ : i.Coprime n), exp (2 * ↑Real.pi * I * (↑i / ↑n)) = ζ

x ∈ rootsOfUnity n ℂ ↔ ∃ i < n, exp (2 * ↑Real.pi * 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.

(primitiveRoots k ℂ).card = k.totient

theorem IsPrimitiveRoot.norm'_eq_one {ζ : ℂ} {n : ℕ} (h : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :

theorem IsPrimitiveRoot.arg_ext {n m : ℕ} {ζ μ : ℂ} (hζ : IsPrimitiveRoot ζ n) (hμ : IsPrimitiveRoot μ m) (hn : n ≠ 0) (hm : m ≠ 0) (h : ζ.arg = μ.arg) :

ζ = μ

ζ.arg = 0 ↔ ζ = 1

theorem IsPrimitiveRoot.arg_eq_pi_iff {n : ℕ} {ζ : ℂ} (hζ : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :

ζ.arg = Real.pi ↔ ζ = -1

theorem IsPrimitiveRoot.arg {n : ℕ} {ζ : ℂ} (h : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :

∃ (i : ℤ), ζ.arg = ↑i / ↑n * (2 * Real.pi) ∧ IsCoprime i ↑n ∧ i.natAbs < n

‖↑ζ‖ = 1