Definition:Root of Unity/Order

Definition

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

Let $F$ be a field.

Let $U_n = \set {z \in F: z^n = 1}$ be the $n$th roots of unity of $F$.

Let $z \in U_n$.

The order of $z$ is the smallest $p \in \Z_{> 0}$ such that:

$z^p = 1$