Definition:Roots of Unity

Definition

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

Let $F$ be a field.

The $n$th roots of unity of $F$ are defined as:

$U_n = \left\{{z \in F: z^n = 1}\right\}$

Complex Roots of Unity

Let $F = \C$.

Then the complex roots of unity are the elements of the set:

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

This follows from Roots of Unity.

Primitive Root of Unity

A primitive $n^\text{th}$ root of unity of $F$ is an element $\alpha \in U_n$ such that:

$U_n = \left\{{1,\alpha, \ldots, \alpha^{n-1}}\right\}$

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 44$