On Quadratic Gauss Sums over Local Fields

Abstract

The classical Gauss sum attached to the Legendre symbol $\left( {\frac{.}{p}} \right)$with an odd prime p is

$$ {g_p} = \frac{1}{{\sqrt p }}\sum\limits_{\mathop {x\bmod p}\limits_{\left( {x,p} \right) = 1} } {\left( {\frac{x}{p}} \right){e^{2\pi ix/p}} = \frac{1}{{\sqrt p }}} \sum\limits_{x\bmod p} {{e^{2\pi i{x^2}/p}}} $$

which is equal to 1 for p≡l (mod 4) and i for p≡3 (mod 4).