Gauss sumGaussSum2002-06-22 04:05:392003-10-18 15:17:19Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}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 {\em 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
$$
g_1(\chi) =
\begin{cases}
\sqrt{p}, & p \equiv 1 \pmod{4}, \\
i \sqrt{p}, & p \equiv 3 \pmod{4}.
\end{cases}
$$
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\bibitem{ir} Kenneth Ireland \& Michael Rosen, {\em A Classical Introduction to Modern Number Theory}, Second Edition, Springer--Verlag, 1990.
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