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Gauss and Eisenstein Sums of Order Twelve

Published:2003-09-01
Printed: Sep 2003
• S. Gurak
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Abstract

Let $q=p^{r}$ with $p$ an odd prime, and $\mathbf{F}_{q}$ denote the finite field of $q$ elements. Let $\Tr\colon\mathbf{F}_{q} \to\mathbf{F}_{p}$ be the usual trace map and set $\zeta_{p} =\exp(2\pi i/p)$. For any positive integer $e$, define the (modified) Gauss sum $g_{r}(e)$ by $$g_{r}(e) =\sum_{x\in \mathbf{F}_{q}}\zeta_{p}^{\Tr x^{e}}$$ 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 ambiguity present in a previous evaluation. Here I generalize Evans' result to give a complete determination of the sum $g_{r}(12)$.
 MSC Classifications: 11L05 - Gauss and Kloosterman sums; generalizations 11T24 - Other character sums and Gauss sums