[[Ring theory MOC]]
# Group of roots of unity

Let $R$ be a [[commutative ring]].
The **roots of unity** 
$$
\begin{align*}
\mu = \{ \zeta \in R : (\exists m \in \mathbb{N}) [\zeta^m = 1] \}
\end{align*}
$$
form a [[subgroup]] of the [[group of units]]. #m/thm/ring

> [!check]- Proof
> Suppose $\zeta,\xi \in \mu$, then these are both units and $(\zeta \xi)^{m_{1}m_{2}} = 1$ where $\zeta^{m_{1}} = \xi^{m_{2}} = 1$. <span class="QED"/>

In particular, if $R$ is the $n$th [[cyclotomic field]], we denote this group by $\mu_{n}$.

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