| Version current |
Version 3 |
| 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 |
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 |
| $$ |
|
| g_a(\chi) := \sum_{t \in \Z/p\Z} \chi(t) e^{2 \pi i a t/p} |
g_a(\chi) := \sum_{t \in \Z/p\Z} \chi(t) e^{2 \pi i a t/p} |
| $$ |
|
| is called a {\em Gauss sum} on $\Z/p\Z$ associated to $\chi$. |
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 |
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 |
| $$ |
|
| g_1(\chi) = |
g_1(\chi) = |
| \begin{cases} |
\begin{cases} |
| \sqrt{p}, & p \equiv 1 \pmod{4}, \\ |
\sqrt{p}, & p \equiv 1 \pmod{4}, \\ |
| i \sqrt{p}, & p \equiv 3 \pmod{4}. |
i \sqrt{p}, & p \equiv 3 \pmod{4}. |
| \end{cases} |
\end{cases} |
| $$ |
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{ir} Kenneth Ireland \& Michael Rosen, {\em A Classical Introduction to Modern Number Theory}, Second Edition, Springer--Verlag, 1990. |
\bibitem{ir} Kenneth Ireland \& Michael Rosen, {\em A Classical Introduction to Modern Number Theory}, Second Edition, Springer--Verlag, 1990. |
| \end{thebibliography} |
\end{thebibliography} |
