In my algebraic number theory lecture we defined Gauss sums as follows. However, I am quite unsure whether this definition is correct. My intuition says "there is a mistake somewhere". I tried double-checking with Pierre Samuels book (§5.5, page 78), but Samuel uses slightly different notation.
If there is indeed a mistake in my defintion, would somebody be so kind and give me a clean (standard) defintion of Gauss sums.
I would think that $\zeta$ should be a primitive $q$-th (not $p$-th) root of unity. In characteristic $p$, the only $p$-th root of unity is unity. That's because $x^p-1=(x-1)^p$ in characteristic $p$; the cross-terms that you'd ordinarily expect have coefficients that are divisible by $p$ and thus vanish in characteristic $p$.