Let $p$ be a prime. Let $\chi$ be any multiplicative group character on $\Z/p\Z$ (that is, any group homomorphism of multiplicative groups $(\Z/p\Z)^\times \to \C^\times$ ). For any $a \in \Z/p\Z$ , the complex number $$ g_a(\chi) := \sum_{t \in \Z/p\Z} \chi(t) e^{2 \pi i a t/p} $$ is called a Gauss sum on $\Z/p\Z$ 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

Bibliography

1

Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory, Second Edition, Springer-Verlag, 1990.