Let $F$ be a finite extension of $K=\mathbb{F}_q$. Prove that for $\alpha \in F$ we have $N_{F/K}(\alpha)=1$ if and only if $\alpha=\beta^{q-1}$ for some $\beta \in F^*$.
I already proved that if $\alpha=\beta^{q-1}$ then the norm is iqual to 1. For the other implication i have this idea $N_{F/K}(\alpha)=\alpha^{\frac{q^m-1}{q-1}}=1$ but i know that $\alpha=\alpha^{q^m}$ because $N_{F/K}(\alpha)= N_{F/K}(\alpha^q)$ then $\alpha^{q^m-1}=1$ so $\alpha^{\frac{q^m-1}{q-1}}=\alpha^{q^m-1}=1$.