Root of unity

Definition

Let $n$ be a positive integer. A complex number $z$ is an $n$ th root of unity when $z^n = 1$ . A complex number $z$ is a root of unity when it is an $n$ th root of unity for some positive integer $n$ .

Remarks

The same definition works in any field, or in any commutative ring. The $n$ th roots of unity in $\mathbb{C}$ form a group under multiplication, written $\mu_n$ . For example, the $4$ th roots of unity are $1$ , $i$ , $-1$ and $-i$ . Each root of unity $z$ satisfies $|z| = 1$ , because $|z|^n = |z^n| = 1$ .

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Complex numbers object
Root of unity definition

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Tag	`0S61N75H45`
URI	`m2://0S61N75H45`
Kind	definition
Writing	W1 stated
MSC	11R18
Defines	Root of unity
Uses	Complex numbers

Root of unity

Definition

Remarks

Reading path

Where this leads

Further reading

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