Given a finite field $\mathbb{F}_{q}$ and $\mathbb{F}_{q^m}$ an extension, define the trace map as $$\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(a):=\sum_{j=0}^{m-1}a^{q^j}.$$
I have managed to prove some properties of this map (like linearity), but I am not convinced that $\text{Tr}(a)\in\mathbb{F}_{q}$ for all $a\in\mathbb{F}_{q^m}$. Also, why is it that, given $c\in\mathbb{F}_{q}$, $$\text{Tr}(a)=c$$ has $q^{m-1}$ distinct solutions? Any help on this? Thank you in advance.
Then you need to prove that $Tr$ is not the zero map, when $p\nmid m$ this is easy,
Otherwise take $a$ such that $\Bbb{F}_{q^m}=\Bbb{F}_q(a)$ and construct the $m\times m$ matrices $A_{ij} = a^{jq^i},B_{ij}=Tr(a^i a^j)$ so that $B=A^\top A$. The rows of $A$ are linearly independent, therefore $\det(A)\ne 0$ and $\det(B)\ne 0$ which implies that $Tr$ is not the zero map.
Being $\Bbb{F}_q$ linear and non-zero $Tr$ is immediately surjective so its kernel has size $q^m / q$.