It is given a set $Z=\{z_i\}_{i=1 \dots n} \subset \mathbb{C}$, I'm interested in finding (the) minimal conditions under which $Z$ contains exactly the $n$-th roots of the unity, that is $Z=\{ e^{\frac{i2k\pi}{n}} \}_{k=0 \dots n-1}$. Although I don't have an answer yet, I made some attempts to see what properties may be useful for the task:
My first attempt
We require that (remember that $Z$ is a set of $n$ complex numbers):
(i) $|z|=1$ for every $z \in Z$.
(ii) $Z$ is closed under conjugation, that is if $z \in Z$, then also $\overline{z} \in Z$.
(iii) $\sum_{z \in Z} z =0$
Unfortunately, this is not enough: consider $Z=\{ e^{i(\frac{\pi}{4}+k\frac{\pi}{2})} \}_{k=0,1,2,3}$, then $Z$ satisfies the conditions, but of course those are not the $4$-th roots of the unity.
A further idea was to also require that $1 \in Z$, and this seems to work, if I'm not missing something. I'm iterested in different perspectives and approaches to this problem: more elegant or sinthetic conditions (provided mine are correct), famous theorems or results concerning this topic. Any contribution is appreciated.