The $n$-th roots of unity

Modulus. $r^n=1$, $r\ge0$, so $r=1$.

Argument. $n\theta\equiv0\pmod{2\pi}$, so $\theta=2k\pi/n$, $k\in\mathbb{Z}$.

Distinct roots. For $k=0,1,\ldots,n-1$, the roots are
$$z_k=e^{2ik\pi/n}.$$
Other values of $k$ give the same roots modulo $n$.

Conclusion. The $n$-th roots of unity are $\{e^{2ik\pi/n}: k=0,\ldots,n-1\}$. They form a cyclic group of order $n$ under multiplication.