Skip to the content

Math 2.0

Group theory and generalizations Object

Group of nth roots of unity

Builds on Abelian group

Description

Let $n$ be a positive integer. The group of $n$ th roots of unity is the set $\mu_n = \{z \in \mathbb{C} : z^n = 1\}$ with complex multiplication. It is a finite abelian group of order $n$ , and its elements are $e^{2\pi i k / n}$ for $k = 0, 1, \ldots, n - 1$ . It is cyclic, and the element $e^{2\pi i / n}$ generates it.

Remarks

The group $\mu_n$ is a subgroup of the circle group. For $n = 4$ , the group is $\{1, i, -1, -i\}$ . A generator of $\mu_n$ is called a primitive $n$ th root of unity, and there are $\varphi(n)$ of them, where $\varphi$ is Euler's totient function.

Reading path

To understand this page from the ground up, read these 5 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

Binary operation definition
Group concept
Commutativity definition
Abelian group definition
Group of nth roots of unity object

Further reading

Web https://en.wikipedia.org/wiki/Root_of_unity, Group of nth roots of unity

About this page

Tag	`0WX11EA3F3`
URI	`m2://0WX11EA3F3`
Aliases	`roots-of-unity`
Kind	object
Writing	W0 stub
MSC	20K01
Instance of	Abelian group
Uses	Abelian group

Markdown · JSON · Edit on GitHub

What this page needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)
field same_as: the identifiers in the databases (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.