Gauss sum in character theory

Question

Define a non trivial additive character $\psi: \mathbb F_q \rightarrow \mathbb C$ s.t $\psi(x)= \xi^{Tr(x)}$ where $\xi=e^{2\pi i /p}$, where $Tr(x):\mathbb F_q \rightarrow \mathbb F_p$ .
First question: is it correct that $Tr(x)= x+ x^p + ... + x^{p^{r-1}}$, with $q=p^r$?
Define now a multiplicative character $\chi:\mathbb F_q^* \rightarrow \mathbb C^*$ and extende it so that $\chi(0)=0$.
Define finally the Gauss sum of $\chi$ as $g(\chi)=\displaystyle \sum_{x \in \mathbb F_q}\chi(x)\psi(x)$.
The goal now is to prove that this sum for the trivial character $\epsilon$ is equal to $-1$.
Clearly $g(\epsilon)=\displaystyle \sum_{x \in \mathbb F_q}\psi(x)$, but I don't know how to calculate this sum.