Definition:Root of Unity/Primitive

Definition

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

Let $F$ be a field.

Let $U_n$ denote the set of all $n$th roots of unity.

Definition 1

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

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

Definition 2

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

$\forall m : 0 < m < n : \alpha^m \ne 1$

Also see

• Equivalence of Definitions of Primitive Root of Unity

• Definition:Primitive Complex Root of Unity

• Results about primitive roots of unity can be found here.