I am trying to figure out the proof of the quadratic reciprocity law in Serre's book A Course in Arithmetic. I think my question is possible to answer independent of the book though:
Let $l$ and $p$ be two distinct prime numbers. Let $\Omega$ be an algebraic closure of $F_p$, and let $w \in \Omega$ be a primitive $l$-th root of unity. At the end of lemma 1 Serre writes that \begin{equation} l-1-\sum\limits_{u\in F_l^*}w^u =l \end{equation}. This suggests that: \begin{equation} \sum\limits_{u\in F_l}w^u=0 \end{equation}
This is supposed to be obvious since Serre just says it like this, but I don't see it! Thank you!