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Let $q$ be a power of prime $p$ and $\zeta_p$ be the complex $p$ th root of unity. We denote by $\mathbb{F}_q$ the finite field of $q$ elements and by $Tr$ the absolute trace function $\mathbb{F}_q\rightarrow \mathbb{F}_p$ . The Kloosterman sum at a point $a \in \mathbb{F}_q$ is defined by the equation

$\hspace{3cm} K_{q}(a)=\displaystyle\sum_{x\in\mathbb{F}_q^*}\psi(x+ax^{-1})$

where $\psi:\mathbb{F}_q\rightarrow \mathbb{Q}(\zeta_p)$ is the canonical additive character of $\mathbb{F}_q$ defined by $\psi(x)=\zeta_p^{Tr(x)}.$

FACT. It is known that $K_{q}(a)\in\mathbb{Z}$ for $p=2,3$.

QUESTION. Does there exist a prime $p\; (\ne 2,3$) such that $K_{q}(a)\in \mathbb{Z} ?$

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    $\begingroup$ B Fisher has studied the Galois action on Kloosterman sums in "Distinctness of Kloosterman sums" (Contemp. Math., vol. 133, pp. 81-102) and "Kloosterman sums as algebraic integers" (Math. Ann. 301, 485--505, 1995). These papers might have the answer to the question. $\endgroup$
    – Denis Chaperon de Lauzières
    Commented Oct 18, 2016 at 18:44
  • $\begingroup$ Thank you @DenisChaperondeLauzières. I have gone through those papers, as for my knowledge Fisher hasn't discussed the above problem. But those papers gave me some useful information. Thank you. $\endgroup$
    – sampath
    Commented Oct 19, 2016 at 6:52

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Kononen, Rinta-Aho, and Vaananen, On integer values of Kloosterman sums, IEEE Transactions on Information Theory 56 (August 2010) 4011-4013, MR2752481 (2011m:11167), discuss the question, and give some examples with $q=25$; if $\alpha$ is a primitive element, then $\alpha^3,\alpha^9,\alpha^{15},\alpha^{21}$ are all said to work.

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  • $\begingroup$ @sam, note that there's no need to @ the poster; he or she will automatically be notified. $\endgroup$
    – LSpice
    Commented Oct 19, 2016 at 21:06

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