# 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 $(\mathbb{Z}/p\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$). 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

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

## References

• 1 Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory, Second Edition, Springer–Verlag, 1990.
Title Gauss sum GaussSum 2013-03-22 12:48:28 2013-03-22 12:48:28 djao (24) djao (24) 7 djao (24) Definition msc 11L05 KloostermanSum