Gauss sum

Let $p$ be a prime. Let $\chi$ be any multiplicative group character on $\mathbb{Z}/p\mathbb{Z}$ (that is, any group homomorphism of multiplicative groups ${(\mathbb{Z}/p\mathbb{Z})}^{\times }\to {ℂ}^{\times }$). For any $a\in \mathbb{Z}/p\mathbb{Z}$, the complex number

$$g_a(\chi ):=\sum _{t\in \mathbb{Z}/p\mathbb{Z}}\chi (t)e^{2\pi iat/p}$$

is called a Gauss sum on $\mathbb{Z}/p\mathbb{Z}$ associated to $\chi$.

In general, the equation $g_a(\chi )=\chi (a^{-1})g_1(\chi )$ (for nontrivial $a$ and $\chi$) reduces the computation of general Gauss sums to that of $g_1(\chi )$. The absolute value of $g_1(\chi )$ is always $\sqrt{p}$ as long as $\chi$ is nontrivial, and if $\chi$ is a quadratic character (that is, $\chi (t)$ is the Legendre symbol $\left(\frac{t}{p}\right)$), then the value of the Gauss sum is known to be