Sum of the roots of unity

Builds on Primitive root of unity Sum of a finite geometric progression

Statement

Let $n \ge 2$ be an integer and let $\zeta$ be a primitive root of unity. Then $\sum_{k=0}^{n-1} \zeta^k = 1 + \zeta + \zeta^2 + \dots + \zeta^{n-1} = 0.$

Remarks

Proof: $\zeta \neq 1$ , so the geometric sum formula gives $\frac{\zeta^n - 1}{\zeta - 1} = 0$ . Thus the sum of all the $n$ th roots of unity is $0$ for $n \ge 2$ . For $n = 1$ the sum is $1$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Complex numbers object
Mathematical induction concept
Root of unity definition
Sum of a finite geometric progression statement
Primitive root of unity definition
Sum of the roots of unity statement

Tag	`02PETK9PQ4`
URI	`m2://02PETK9PQ4`
Kind	statement, proposition
Writing	W0 stub
MSC	11R18
Status	This copy of the site has no build data. Run `m2 status 02PETK9PQ4`.
Uses	Primitive root of unity, Sum of a finite geometric progression

Sum of the roots of unity

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here