How do you characterize all the linear relations satisfied by $n$th roots of unity with real, integral and non-negative integral coefficients?
Here are two examples for 3rd and 4th root:
Let $\omega_{i}$ be the primitive $i$th root of unity, then
For $2$nd root, $1 + \omega_{2} = 0$;
For $3$rd root, $1+ \omega_{3} + \omega_{3}^{2}=0$