Let $F_q$ and $F_{q^{\ell}}$ be the finite fields with $q, q^{\ell}$ elements respectively, where $\ell \ge 3$ is a prime and $\gcd(\ell, q)=1$. I have the following question:
Does there exist $\ell, q\;$ (as above) and an element $\alpha \in F_{q^{\ell}}$ such that $Tr_{F_{q^{\ell}}/F_q}(\alpha)= 0$ and $N_{F_{q^{\ell}}/F_q}(\alpha)=1$ ?