# Gamma function and Gauss sums

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?

• The Gamma function is defined on the complex numbers, and they form an infinite field. – Gerry Myerson Jul 28 '14 at 12:54

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.