Let $p$ be prime, and $z\_0, z\_1, ..., z\_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$


Is it true or false that a combination of two (or more, in general) of the roots can give us another root of the same order? 


In mathematical terms, does there exist indices $i\_1,i\_2,...,i\_s, j,$
such that
$z\_j = \sum\_{k=1}^s z\_{i\_k}?$


It seems to be that this is not possible, but I also don't have proof of that.


Thank you!



Can the sum of two roots of unity be a root of unity?
If $p$ is intended to be prime, you should say so explicitly. I'll assume $p$ is prime. Then the subset sums are distinct except that the sum of all $p$th roots of unity is $0$, the sum over the empty set. Any coincidence of subset sums $\sum\_{i \in I} \zeta\_p^i = \sum\_{j\in J} \zeta\_p^j $produces a polynomial of degree at most $p-1$ with coefficients in $\lbrace-1,0,1\rbrace$ so that $\zeta\_p$ is a root. This polynomial must be a multiple of the minimum polynomial for $\zeta\_p$, the $p$th cyclotomic polynomial $\Phi\_p(x) =1 + x + ... + x^{p-1}$. The only possibilities are scalar multiples corresponding to $1 + \zeta\_p + ... + \zeta\_p^{p-1} = 0$.


