# The sum of unity roots

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?

• The standard result is that $\eta-1$ generates the unique prime ideal above $p$ in $\Bbb Z[\eta]$. – Angina Seng Jul 15 '17 at 5:21
• And a quick way to see that is to see look at the norm of $\eta - 1$ down to $\mathbb{Q}$. The norm is $\prod_{i=1}^{p-1} (\eta^i - 1)$, and which is $\prod_{i=1}^{p-1} (\eta^i x^i - 1) = x^{p-1} + x^{p-2} + ... + x + 1$ evaluated at $1$, so it equals $p$. Since it has prime norm, the absolute norm of the ideal $(\eta - 1)$ is prime, so quotienting the ring of integers by this ideal gives a field, so $(\eta - 1)$ is a prime ideal (and we see it divides $p$). – Barry Smith Jul 15 '17 at 11:16