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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 ?

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

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