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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 $g(\chi_{1})g(\chi_{2})\cdots g(\chi_{n})=J(\chi_{1},\cdots,\chi_{n})g(\prod_{i=1}^{n}\chi_{i})$

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If you're stuck, how do you know the relation between Gauss and Jacobi sums is relevant? That remark makes it sound like this is a homework problem. – KCd Feb 23 '12 at 20:55
up vote 2 down vote accepted

Hint: consider $$\sum_{t=0}^{p-1}\sum_{x_i}\sum_{x_2}\cdots\sum_{x_n}e^{2\pi it(x_1^2+x_2^2+\cdots+x_n^2)/p}$$ The sums on the $x_i$ all run from $0$ to $p-1$.

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