Enumerable partition set $C/R$

Question

Let $C=\{z\in\mathbb{C}:|z|=1\}$ and let $R$ be an equivalence relation such that $zRw$ if $\exists n\ge1$ such that $z^n=w^n$. Is $C/R$ enumerable? I found that C is not enumerable because it is in bijection with $[0,2\pi)\subseteq \mathbb{R}$ but for the partition set I can't find I way to show its non-enumerability.

Answer 1

Just show that for each $z$, its class $[z]$ is countable. Observe that $[z]= \bigcup\limits_{n \in \mathbb{N}}\{x \in C \ | \ x^n - z^n =0\}$. Thus, for each $n$ there is at most $n$ values $x$ that satisfies the polynomial $x^n - z^n =0$, so the class of $z$ in countable. We know that $C=\bigcup[z]$, so we must have an uncountable number of classes, otherwise $C$ would be countable.