PlanetMath: Gauss sum

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Gauss sum

(Definition)

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

$\displaystyle g_a(\chi) := \sum_{t \in \mathbb{Z}/p\mathbb{Z}} \chi(t) e^{2 \pi i a t/p}$

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 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

$\begin{displaymath} g_1(\chi) = \begin{cases} \sqrt{p}, & p \equiv 1 \pmod{4}, \ i \sqrt{p}, & p \equiv 3 \pmod{4}. \end{cases}\end{displaymath}$

Bibliography

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

"Gauss sum" is owned by djao.

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