> [!NOTE] Definition ($n$th roots of unity) > Let $n\in \mathbb{N}.$ The $n$th roots of unity is the set $U_{n}=\{ z\in \mathbb{C}: z^{n}=1 \}$ **Notation**: $U_{n}$ but not standard notation. # Properties By [[Roots of Unity under Multiplication form Cyclic Group]], $(U_{n},\times)$ is a cyclic group. Note that its generators are known as the [[Primitive Root of Unity|primitive nth roots of unity]]. By [[Roots of Unity is a Subgroup of Unit Group of Complex Numbers]], $U_{n}$ is a subgroup of $\mathbb{C}^{*}.$