= Quadratic finite-field norm and trace surjectivity
{title2=$N(a)=a^{q+1},\quad\operatorname{Tr}(a)=a+a^q$}
For $\mathbb F_{q^2}/\mathbb F_q$, the <field norm> maps the nonzero elements onto $\mathbb F_q^*$: 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 $a\mapsto\operatorname{Tr}(ac)$ is onto whenever $c\ne0$.
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