CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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For $n\ge2$ , the sum of all $n$ -th roots of unity is zero

Statement

For $n\in\mathbb N$ with $n\ge2$ , the sum of all $n$ th roots of unity is $0$ . The conventions and prerequisite facts used below are recorded in The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n\ge1$ , The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$ , Integer powers in the complex field, The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ( $n = 0$ ) equal to the identity, Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either.

Facts & Assumptions

Given: A natural $n\ge2$ and $\zeta=\exp\!\left(i\frac{2\pi}{\iota_{\mathbb R}(n)}\right)$ , with $\iota_{\mathbb R}$ as in The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n\ge1$ .

Proof

technique · direct

1.1

The roots are the distinct list $1,\zeta,\ldots,\zeta^{n-1}$ , with $\zeta^n=1$ and $\zeta\ne1$ .

given

1.2

The cyclic successor map on the initial segment $n$ is a permutation; the commutative-monoid permutation rule therefore gives $\zeta S=S$ for $S=\sum_{k<n}\zeta^k$ .

given

2.1

Thus $(\zeta-1)S=0$ , and field cancellation gives $S=0$ .

algebra∎

Depends on

Used by

The five fifth roots of unity and their sum Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 82 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

For n≥2n\ge2n≥2, the sum of all nnn-th roots of unity is zero