Let $\mathbb{F}_{q^m}$ be an extension field of the finite field $\mathbb{F}_q$. Let $T := Tr_{\mathbb{F_{q^m}}|\mathbb{F}_q}$ be the trace map and $\gamma$ a primitive element of $\mathbb{F}_{q^m}$.
Prove that for ever integer $k$ the elements $\gamma^k$, $\gamma^{k+1}$, ... ,$\gamma^{k+m-1}$ are linearly independent over $\mathbb{F}_q$. Moreover, show that there exists an integer $0 \leq k \leq q^{m-2}$ such that $T(\gamma^k) = T(\gamma^{k+1}) = ... = T(\gamma^{k+m-1}) = 0$.
I am already stuck with the first part. I have to show that if
$c_0\gamma + c_1\gamma^{k+1}+ ... +c_{m-1}\gamma^{k+m-1} = 0 $ for $c_i \in \mathbb{F}_q$ then all $c_i = 0$
but I don't know how to do this.
Can anyone give me a hint? Thanks!