Thread: Sum of powers of unity root

1. Sum of powers of unity root

Let $\displaystyle G_n(1)=\{q\in Z_n|(q,n)=1\}$ $\displaystyle S\subset G_n (1)$ (strict subset). Prove that:
$\displaystyle \sum_{q \in S}\omega^q_n\notin Z$
where $\displaystyle \omega_n=e^{i\frac{2\pi}n}.$