Gauss and Eisenstein Sums of Order Twelve

Canadian mathematical bulletin

关键词: Jacobian Conjecture;    injectivity;    Monge--Ampère equation;   

DOI  :  10.4153/CMB-2003-036-9

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $q=p^{r}$ with $p$ an odd prime, and $mathbf{F}_{q}$ denote the finitefield of $q$ elements. Let $Trcolonmathbf{F}_{q} omathbf{F}_{p} $ bethe usual trace map and set $zeta_{p} =exp(2pi i/p)$. For any positiveinteger $e$, define the (modified) Gauss sum $g_{r}(e)$ by$$g_{r}(e) =sum_{xin mathbf{F}_{q}}zeta_{p}^{Tr x^{e}}$$Recently, Evans gave an elegant determination of $g_{1}(12)$ in terms of$g_{1}(3)$, $g_{1}(4)$ and $g_{1}(6)$ which resolved a sign ambiguitypresent in a previous evaluation. Here I generalize Evans' result to givea complete determination of the sum $g_{r}(12)$.

【 授权许可】

Unknown