Let $p\not=2$ be a prime. Suppose that $\zeta_1,\cdots,\zeta_n$ are $p$-th unity roots. If their sum $S$ is an integer, show that $S$ is congruent to $n$ modulo $p$.
I don't know how to deal with such problems. Generally, is it necessary that $\zeta_i-1\in pO$, where $O$ is the algebraic integer ring of $\mathbb{Q}(\xi)$ and $\xi$ is a primitive $p$-th root?