Definition:Root of Unity/Complex

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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:


Complex-6th-Roots-of-1.png


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

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


Sources