This question is a bit of a follow up to this question.

Let us consider the finite field $\mathbb{F}_q$ and its algebraic closure $\mathbb{F}$, viewed as an additive abelian group. Its group of linear characters, ${\rm Hom}(\mathbb{F},\mathbb{C}^*)$, consists of the group homomorphisms from the (additive) abelian group $\mathbb{F}$ to the multiplicative group of $\mathbb{C}\setminus \{0\}$. If $\lambda$ is linear character that is fixed by the Frobenius morphism ($x\mapsto x^q$), then is it true that $\lambda $ trivial? (Possibly because the Lang map is surjective?)

And is there a good description of the linear characters? And a good relation with ${\rm Hom}(\mathbb{F}_q,\mathbb{C}^*)$ commuting with the trace maps?