In this Wikipedia article appears this : "Gauss sums are the analogues for finite fields of the Gamma function."
What was the relation between gamma functions and non-finite fields?
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1The Gamma function is defined on the complex numbers, and they form an infinite field. – Gerry Myerson Jul 28 '14 at 12:54
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1A very good overview is given in the book by Lemmermeyer, Reciprocity laws, from Euler to Eisenstein, p. 139. See also https://math.stackexchange.com/questions/523656. – Watson Jul 24 '20 at 14:25
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Gauss sums are closely related to an analogue of the complex Gamma function - the $p$-adic Gamma function. Let $p$ be an odd prime. The $p$-adic Gamma function is defined by $$ \Gamma_p(z)=\lim_{m\to z}(-1)^m\prod_{0<j<m, (p,j)=1}j, $$ where $m$ approaches $z\in \mathbb{Z}_p$ through positive integers. There is a direct connection between the $p$-adic Gamma function and Gauss sums, see the article of Gross and Koblitz.
Dietrich Burde
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