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 ?

  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.