Complex roots of unity #

In this file we show that the n-th complex roots of unity are exactly the complex numbers e ^ (2 * real.pi * complex.I * (i / n)) for i ∈ finset.range n.

Main declarations #

complex.mem_roots_of_unity: the complex n-th roots of unity are exactly the complex numbers of the form e ^ (2 * real.pi * complex.I * (i / n)) for some i < n.
complex.card_roots_of_unity: the number of n-th roots of unity is exactly n.

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theorem complex.is_primitive_root_exp_of_coprime (i n : ℕ) (h0 : n ≠ 0) (hi : i.coprime n) :

is_primitive_root (complex.exp (((2 * ↑π) * complex.I) * (↑i / ↑n))) n

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theorem complex.is_primitive_root_exp (n : ℕ) (h0 : n ≠ 0) :

is_primitive_root (complex.exp ((2 * ↑π) * complex.I / ↑n)) n

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theorem complex.is_primitive_root_iff (ζ : ℂ) (n : ℕ) (hn : n ≠ 0) :

is_primitive_root ζ n ↔ ∃ (i : ℕ) (H : i < n) (hi : i.coprime n), complex.exp (((2 * ↑π) * complex.I) * (↑i / ↑n)) = ζ

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theorem complex.mem_roots_of_unity (n : ℕ+) (x : units ℂ) :

x ∈ roots_of_unity n ℂ ↔ ∃ (i : ℕ) (H : i < ↑n), complex.exp (((2 * ↑π) * complex.I) * (↑i / ↑n)) = ↑x

The complex n-th roots of unity are exactly the complex numbers of the form e ^ (2 * real.pi * complex.I * (i / n)) for some i < n.

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theorem complex.card_roots_of_unity (n : ℕ+) :

fintype.card ↥(roots_of_unity n ℂ) = ↑n

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theorem complex.card_primitive_roots (k : ℕ) (h : k ≠ 0) :

(primitive_roots k ℂ).card = k.totient