This question is a bit of a follow up to <a href="https://mathoverflow.net/questions/283965/representations-of-algebraic-groups">*this*</a> 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?