I am stuck with the following. I want to find the number of solution to $\sum_{i=1}^{n} x_{i}^{2}=0$ in the finite field $\mathbb{F_{p}}$.
The following relation between Gauss sums and Jacobi sums is relevant: Lemma: If $\chi_{1},\cdots,\chi_{n}$ are nontrivial characters and $\prod_{i=1}^{n}\chi_{i} $ is also nontrivial, then $g(\chi_{1})g(\chi_{2})\cdots g(\chi_{n})=J(\chi_{1},\cdots,\chi_{n})g(\prod_{i=1}^{n}\chi_{i})$