Trace and cyclotomic field

Question

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.