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Let $k=\mathbb{F}_q$ be the finite field with $q=p^r$ elements. Assume that $q$ is odd. Let $\psi$ be a nontrivial additive character of $k$. For $a,b\in k^\times$, consider the Gauss sum $$G(a,b)=\sum_{x\in k^\times}\psi\left(ax+\frac{b}{x}\right).$$ Is there a general formula for $G(a,b)$ in terms of $a,b$?

Any hints, references are appreciated.

Q-Zhang
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  • Checking for small values of $p$, I find that this is likely to be some "random" element of the totally real subfield of $\Bbb Q(\zeta_p)$. Of course you may rewrite the sum in different forms, but I'm afraid there is no satisfactory result. – WhatsUp Mar 19 '20 at 23:13
  • @WhatsUp Thanks for your comment. I also tested several small values and could not guess a nice formula. I am just wondering if those $G(a,b)$ has some known nice properties, even there is no general formula for their exact values. – Q-Zhang Mar 20 '20 at 03:17

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