Definition:Root of Unity/Complex/Primitive

Definition

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

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

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

Equivalently, an $n$th root of unity is primitive if and only if its order is $n$.

The primitive complex cube roots of unity are:

$\ds \omega$

$=$

$\, \ds e^{2 \pi i / 3} \, $

$\, \ds = \, $

$\ds -\frac 1 2 + \frac {i \sqrt 3} 2$

$\ds \omega^2$

$=$

$\, \ds e^{4 \pi i / 3} \, $

$\, \ds = \, $

$\ds -\frac 1 2 - \frac {i \sqrt 3} 2$