> [!NOTE] Lemma > Let $n\in\mathbb{N}^{+}.$ Let $U_{n}$ denote the [[Roots of Unity|nth roots of unity]]. $U_{n}$ is a [[Subgroup|subgroup]] of $\mathbb{C}^{*}$ where $\mathbb{C}^{*}$ denotes the [[Unit Group of Ring|unit group]] of [[Complex Numbers|ring of complex numbers]]. **Proof**: Clearly $U_{n}\subset C^{*}$ and $1\in U_{n}.$ Suppose $a,b\in U_{n}.$ Then $(ab)^{n}= a^{n}b^{n}=1.$ So $ab\in U_{n}.$ By [[Inverse of Power of Group Element]], $(a^{-1})^{n}=(a^{n})^{-1}=1$ thus $a^{-1}\in U_{n}.$ Thus by [[Two-Step Subgroup Test]], $U_{n}$ is indeed a subgroup of $\mathbb{C}^{*}.$