Let $q=p^k$ be a prime power, let $m\in\mathbb{N}$ and consider the map $$ \tau:\mathbb{F}_{q^m}\longrightarrow\mathbb{F}_{q^m},\quad a\longmapsto\sum_{i=0}^{m-1} a^{q^i}. $$ Show that $\tau$ is not the zero map.
I was able to show that this map is $\mathbb{F}_{q}$-linear, and that $\text{im }\tau\subset\mathbb{F}_q$. However, I am stuck on showing this map is not the zero map. I suppose I could look at $\tau(1)$ but if $m=p$, then $\tau(1)=0$ since the characteristic of $\mathbb{F}_{q^m}$ is $p$. Do I have to handle this as a special case, or should I do something else? Thanks.