# Doubt about $p$-th root of unity

Let $p\in\mathbb{Z}$ be an odd prime. Let $\zeta_p$ be a primitive $p$-th root of unity. Let $K=\mathbb{Q}(\zeta_p)$. Let $O_K$ be the ring of algebraic integrs. I am trying to prove that for any $2\leq i\leq p-1$ $$1+\zeta_p+\dots+\zeta_p^{i-1}\notin \langle1-\zeta_p\rangle$$ in $O_K$. How to prove this ?

• Just take quotients mod (1 - zeta). Then your sum is congruent to i mod p, which is not null. – nguyen quang do Nov 17 '16 at 14:17