Value of $ \sum_{t \in \mathbb{F}_{p^n}} \zeta ^{Tr (t)} $

Question

Let $\zeta = e^{2\pi i/p } $. Then is it true that $ \sum_{t \in \mathbb{F}_{p^n}} \zeta ^{Tr (t)} = 0 $, where $Tr$ is the map $ Tr:\mathbb{F}_{p^n} \rightarrow \mathbb{F}_{p}$ defined by $Tr(a) = a+a^p+a^{p^2}+......a^{p^{n-1}}$? If so, why? I know that $Tr$ is onto.

Answer 1

Look at $$\sum_{l=1}^{p-1} \sum_{t \in \mathbb{F}_{p^n}} \zeta^{l\ Tr (t)}$$ Equivalently show that since $Tr$ is a surjective group homomorphism then $|Tr^{-1}(t)| = p^{n-1}$.