Quadratic finite-field norm and trace surjectivity ($N(a)=a^{q+1},\mathrm{Tr}(a)=a+a^q$)

For ${\mathbb{F}}_{q^2}/{\mathbb{F}}_q$, the field norm maps the nonzero elements onto ${\mathbb{F}}_q^{\ast }$: the multiplicative group of a finite field is cyclic, and the power $q+1$ has image of order $q-1$. The field trace is also onto. Indeed it is an ${\mathbb{F}}_q$-linear map into ${\mathbb{F}}_q$, and is not zero because its polynomial $X+X^q$ has degree $q$ and cannot vanish at all $q^2$ field elements. Consequently $a\mapsto \mathrm{Tr}(ac)$ is onto whenever $c\ne 0$.