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The connection between Gauss sums and the $p$-adic Gamma function is a very deep theorem, and is known as given by the Gross-Koblitz formula.

Here is the original paper by Gross and Koblitz: Gauss sums and the $p$-adic $\Gamma$-function. (1979)

For some more specific cases and applications, see the book by Berndt, Evans, and Williams, "Gauss and Jacobi Sums. (1998)"

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The connection between Gauss sums and the $p$-adic Gamma function is a very deep theorem, and is known as the Gross-Koblitz formula: http://en.wikipedia.org/wiki/Gross%E2%80%93Koblitz_formula

I hope that .

Here is what you were looking for,the original paper by Gross and Koblitz: Gauss sums and the $p$-adic $\Gamma$-function. (1979)

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The connection between Gauss sums and the $p$-adic Gamma function is a very deep theorem, and is known as the Gross-Koblitz formula: http://en.wikipedia.org/wiki/Gross%E2%80%93Koblitz_formula

I hope that is what you were looking for,