Let $K/F$ be an extension of finite field. Show that the norm map $N_{K/F}$ is surjective.
Here is what I have so far:
Since $F$ is a finite field and $K/F$ is a finite extension of degree $n$, so $\operatorname{Gal}(K/F)=\langle\sigma\rangle$, where $\sigma(a)= a^{q}$ with $q=p^{m}=|F|$. In addition, by primitive element theorem, $K=F(\alpha)$ for some $\alpha \in K$.
We want to show $N_{K/F}(\alpha)$ generates $F$. By the definition of norm, we have $N_{K/F}(\alpha)=\alpha^{1+q+\cdots+q^{n-1}}$ and since $(1+q+\cdots+q^{n-1})(q-1)=q^n-1$, we have the order of $N_{K/F}(\alpha)$ is divided by $q-1$.
But I want to show $o(N_{K/F}(\alpha))=q-1$ in order to conclude surjectivity. So any hint for how to proceed? Any other methods are also perferred.