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 ${(\mathbb{Z}/p\mathbb{Z})}^{\times}\to {\u2102}^{\times}$). 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
$${g}_{1}(\chi )=\{\begin{array}{cc}\sqrt{p},\hfill & p\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(mod4),\hfill \\ i\sqrt{p},\hfill & p\equiv 3\phantom{\rule{veryverythickmathspace}{0ex}}(mod4).\hfill \end{array}$$ |
References
- 1 Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory^{}, Second Edition, Springer–Verlag, 1990.
Title | Gauss sum |
---|---|
Canonical name | GaussSum |
Date of creation | 2013-03-22 12:48:28 |
Last modified on | 2013-03-22 12:48:28 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 7 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 11L05 |
Related topic | KloostermanSum |