exp(2 pi i/n) is a primitive root of unity

Builds on Primitive root of unity The nth roots of unity

Statement

Let $n$ be a positive integer. Then $e^{2 \pi i / n}$ is a primitive root of unity.

Remarks

Proof: for $1 \le k < n$ , the number $\zeta^k = e^{2 \pi i k / n}$ has argument $2 \pi k / n$ , which is strictly between $0$ and $2\pi$ . Thus $\zeta^k \neq 1$ . For $n = 4$ this number is $i$ , and for $n = 2$ it is $-1$ .

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Complex numbers object
Real numbers object
Set bounded above definition
Euler's formula statement
Product of exponentials statement
Real and imaginary parts definition
Root of unity definition
Supremum definition
de Moivre's formula statement
Completeness axiom statement
Primitive root of unity definition
Existence of square roots of nonnegative reals statement
Square root definition
Modulus of a complex number definition
Argument of a complex number definition
Principal value of the argument statement
Polar form statement
The nth roots of unity statement
exp(2 pi i/n) is a primitive root of unity statement

Tag	`0CF0XGYHNJ`
URI	`m2://0CF0XGYHNJ`
Kind	statement, proposition
Writing	W0 stub
MSC	11R18
Status	This copy of the site has no build data. Run `m2 status 0CF0XGYHNJ`.
Uses	Primitive root of unity, The nth roots of unity

exp(2 pi i/n) is a primitive root of unity

Statement

Remarks

Reading path

Further reading

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