# Definition:Root of Unity/Complex

## Contents

## Definition

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

The **complex $n$th roots of unity** are the elements of the set:

- $U_n = \set {z \in \C: z^n = 1}$

### Primitive Root of Unity

A **primitive (complex) $n$th root of unity** is an element $\alpha \in U_n$ such that:

- $U_n = \set {1, \alpha, \alpha^2, \ldots, \alpha^{n - 1} }$

### First Root of Unity

The root $e^{2 i \pi / n}$ is known as the **first (complex) $n$th root of unity**.

### Order of Root of Unity

Let $z \in U_n$.

The **order** of $z$ is the smallest $p \in \Z_{> 0}$ such that:

- $z^p = 1$

## Illustration

From Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle the complete set of complex $6$th roots of unity can be depicted as the vertices of the following regular hexagon:

where $\alpha$ is used to denote the first complex $6$th root of unity.

## Examples

### Complex Cube Roots of Unity

The **complex cube roots of unity** are the elements of the set:

- $U_3 = \set {z \in \C: z^3 = 1}$

They are:

\(\displaystyle \) | \(\) | \(\, \displaystyle e^{0 i \pi / 3} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle 1\) | |||||||||

\(\displaystyle \omega\) | \(=\) | \(\, \displaystyle e^{2 i \pi / 3} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle -\frac 1 2 + \frac {i \sqrt 3} 2\) | |||||||||

\(\displaystyle \omega^2\) | \(=\) | \(\, \displaystyle e^{4 i \pi / 3} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle -\frac 1 2 - \frac {i \sqrt 3} 2\) |

The notation $\omega$ for, specifically, the complex **cube** roots of unity is conventional.

### Complex $4$th Roots of Unity

The **complex $4$th roots of unity** are the elements of the set:

- $U_n = \set {z \in \C: z^4 = 1}$

They are:

\(\displaystyle e^{0 i \pi / 4}\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle e^{i \pi / 2}\) | \(=\) | \(\displaystyle i\) | |||||||||||

\(\displaystyle e^{i \pi}\) | \(=\) | \(\displaystyle -1\) | |||||||||||

\(\displaystyle e^{3 i \pi / 2}\) | \(=\) | \(\displaystyle -i\) |

### Complex $5$th Roots of Unity

The **complex $5$th roots of unity** are the elements of the set:

- $U_n = \set {z \in \C: z^5 = 1}$

They are:

\(\displaystyle e^{0 \pi / 5}\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle e^{2 \pi / 5}\) | \(=\) | \(\displaystyle \dfrac {\sqrt 5 - 1} 4 + i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\) | |||||||||||

\(\displaystyle e^{4 \pi / 5}\) | \(=\) | \(\displaystyle -\dfrac {1 + \sqrt 5} 4 + i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}\) | |||||||||||

\(\displaystyle e^{6 \pi / 5}\) | \(=\) | \(\displaystyle -\dfrac {1 + \sqrt 5} 4 - i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}\) | |||||||||||

\(\displaystyle e^{8 \pi / 5}\) | \(=\) | \(\displaystyle \dfrac {\sqrt 5 - 1} 4 - i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\) |

### Complex $6$th Roots of Unity

The **complex $6$th roots of unity** are the elements of the set:

- $U_n = \set {z \in \C: z^6 = 1}$

They are:

\(\displaystyle e^{0 i \pi / 6}\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle e^{i \pi / 3}\) | \(=\) | \(\displaystyle \frac 1 2 + \frac {i \sqrt 3} 2\) | |||||||||||

\(\displaystyle e^{2 i \pi / 3}\) | \(=\) | \(\displaystyle -\frac 1 2 + \frac {i \sqrt 3} 2\) | |||||||||||

\(\displaystyle e^{i \pi}\) | \(=\) | \(\displaystyle -1\) | |||||||||||

\(\displaystyle e^{4 i \pi / 3}\) | \(=\) | \(\displaystyle -\frac 1 2 - \frac {i \sqrt 3} 2\) | |||||||||||

\(\displaystyle e^{5 i \pi / 3}\) | \(=\) | \(\displaystyle \frac 1 2 - \frac {i \sqrt 3} 2\) |

## Also see

- Complex Roots of Unity in Exponential Form, where it is shown that $U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$

- Results about
**the complex roots of unity**can be found here.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 3$. Roots of Unity - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Introduction - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 44$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The $n$th Roots of Unity