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For $n \geq 2$ , the sum of all $n$ -th roots of unity is zero

Statement

For $n \in N$ with $n \geq 2$ , 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 $n \geq 1$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Integer powers in the complex field, The product $g_{0} g_{1} \dots g_{n - 1}$ 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 $n \geq 2$ and $ζ = \exp ⁣ (i \frac{2 π}{ι_{R} (n)})$ , with $ι_{R}$ as in The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$ .

Proof

technique · direct

1.1

The roots are the distinct list $1, ζ, \dots, ζ^{n - 1}$ , with $ζ^{n} = 1$ and $ζ \neq 1$ .

given

1.2

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

given

2.1

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

algebra∎

Depends on

Used by

The disc algebra is unital and separating but not self-adjoint or dense Counterexample
The five fifth roots of unity and their sum Example

Dependency tree · two levels

34 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

For n≥2n≥2, the sum of all nn-th roots of unity is zero