Definition:Root of Unity/Complex/Primitive

Definition

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

Let $U_n$ denote the complex $n$th roots of unity:

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

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 $n$.

Examples

Primitive Complex Cube Roots of Unity

The primitive complex cube roots of unity are:

\(\ds \omega\)

\(=\)

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

\(\, \ds = \, \)

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

\(\ds \omega^2\)

\(=\)

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

\(\, \ds = \, \)

\(\ds -\frac 1 2 - \frac {i \sqrt 3} 2\)

Also see

• Results about the primitive complex $n$th 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): Chapter $2$: Subgroups and Cosets: $\S 44$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): primitive root
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): primitive root