TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-02

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The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$

Statement

Write $ι_{R} : N \to R$ for the canonical-natural map of The canonical natural $ι (n) = n \cdot 1_{F}$ of a field. If $n \in N$ with $n \geq 1$ and $z = r e^{i θ} \neq 0$ , its $n$ distinct roots are $r^{1 / n} \exp ⁣ (i \frac{θ + 2 π ι_{R} (k)}{ι_{R} (n)}), k \in N, 0 \leq k < n .$ For $z = 0$ , the only $n$ th root is $0$ . Thus the $n$ th roots of unity are precisely $\exp ⁣ (i \frac{2 π ι_{R} (k)}{ι_{R} (n)})$ for $k \in N$ with $0 \leq k < n$ . The conventions and prerequisite facts used below are recorded in Every nonzero complex number has a unique polar form $r (\cos θ + i \sin θ)$ with $r > 0$ and $- π < θ \leq π$ , Complex de Moivre formula for every integer exponent, $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ , Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ , The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, Integer powers in the complex field.

Facts & Assumptions

Given: $n \in N$ with $n \geq 1$ and $z \in C$ .

Proof

technique · constructive

1.1

For $z \neq 0$ , construct each listed candidate from the positive real $n$ th root of $r$ ; de Moivre verifies it.

construct

1.2

The kernel theorem shows two listed candidates coincide only when their indices are equal modulo $n$ .

given

2.1

Conversely polar form and the same kernel calculation force every root onto the list; the $z = 0$ case is immediate.

discharge-construct∎

Depends on

Used by

Dependency tree · two levels

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Sources

J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)

The nn-th roots of a complex number and the nn distinct roots of unity for every n≥1n≥1