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

Let us consider the finite field $\mathbb{F}_q$ and it's algebraic closure $\mathbb{F}$, we can consider it as an abelian group, and consider its linear characters $Hom(\mathbb{F},\mathbb{C}^*)$, this is group morphism from the abelian group $\mathbb{F}$ (to the sum) to the abelian multiplicative group of $\mathbb{C}$ , and . If we have a linear character $\lambda$ that is fixed by the Frobenius morphism ($x\mapsto x^q$) then it's true that it's trivial? (Because the Lang map is surjective)

And exist a good description of the linear characters? And a good relation with the $Hom(\mathbb{F}_q,\mathbb{C}^*)$ in a way that the relation commutes with the trace maps.