Let $ψ: \mathbb{F_p} \to Z_p$ with the property $ψ(a+b)=ψ(a)ψ(b)$ where $Z_p$ denotes the p-adic integers. Assume further that $ψ$ is not trivial.
I'm trying to follow my professor's work, but I believe he may have made a typo.
For each $0 \leq k \leq p-2$, define the Gauss sum $g_k= \sum _{x \in \mathbb{F_p}}t^{p-1-k}ψ(x)$.
Let $\sum _k G(k)t^k$ be a polynomial of degree $p-1$ such that $ψ(\overline{t})= \sum _{k=0} ^{p-1}G(k)t^k$ for all $t\in \mathbb{F_p}$
Then, he claims that $G(0)=1$ (which is easy for me to see) but also $G(p-1)=-p/(p-1)$ (which I'm not sure how to prove.)
He also claims that, for $1 \leq k \leq p-2$, $G(k)=g_k/(p-1).$ I believe that this is inconsistent with his previous definition of $g_k$ (after all, shouldn't G(k) be a coefficient instead of a polynomial?) which is why I think he made a typo. Can anyone see an easy way to correct it?
Try $g_k:= \sum _{x \in \mathbb{F}_p}T(x)^{p-1-k}ψ(x)$