Definition:Root of Unity/Complex/First

Definition

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

In Complex Roots of Unity in Exponential Form it is shown that the complex $n$th roots of unity are the elements of the set:

$U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$

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

Notation

The first $n$th root of unity is usually assigned a letter of the Greek alphabet: $\alpha$, $\epsilon$, $\omega$.

$\mathsf{Pr} \infty \mathsf{fWiki}$ intends to use $\alpha$ as standard.

Many texts reserve $\omega$ for the first complex cube root of unity:

$\omega = -\dfrac 1 2 + i \dfrac {\sqrt 3} 2$

which convention is adopted by $\mathsf{Pr} \infty \mathsf{fWiki}$.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 44$