# Trace and cyclotomic field

Let $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field of $p$th roots of unity for the prime $p$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Let $\zeta$ denote any $p$th root of unity. Please show that $\sum_{\sigma\in G}\sigma(\zeta)$ is $-1$ or $p-1$ depending on whether $\zeta$ is or is not a primitive $p$th root of unity.

• If $\zeta$ is not primitive then it has to be 1 since $p$ is prime. So we get $\phi(p)=p-1$ in this case. If $\zeta$ is primitive then $\sigma_k(\zeta)=\zeta^k$ where $\sigma_k\in G$.So $\zeta+...+\zeta^{p-1}=-1$. – user70520 Apr 6 '13 at 22:21
• So...why did you ask the question if you know the answer, @user70520 ? – DonAntonio Apr 7 '13 at 3:29
• I was looking for another way of doing it. – user70520 Apr 7 '13 at 3:56