This question is a bit of a follow up to [*this*](https://mathoverflow.net/questions/283965/representations-of-algebraic-groups) 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?



Linear characters of algebraic closed fields
$\newcommand{\IF}{\mathbb{F}}\newcommand{\IC}{\mathbb{C}}$
For simplicity I'm going to assume $q=p$.


Since all elements in $\overline{\IF\_p}$ have order $p$, any group homomorphism $\overline{\IF\_p}\to\IC^\times$ has either image contained in the group of $p$-th roots of unity $\mu\_p$. In other words we can equivalently ask for homomorphisms of additive groups $\overline{\IF\_p}\to\mathbb{Z}/p$, i.e. $\IF\_p$-vector space homomorphisms $\overline{\IF\_p}\to\IF\_p$.


Let $F$ be the Frobenius automorphism. Since $F-1: \overline{\IF\_p}\to\overline{\IF\_p}$ is surjective, we have $\phi\circ F=\phi \iff 0=\phi\circ(F-1) \iff \phi = 0$.


The next best thing is the normalised galois trace $\tau: \bigcup\_{(p,m)=1} \IF\_{p^m} \to \IF\_p$. That is the unique (up to multiplication with a constant) $F$-invariant linear form on that field.


Every finite field has a one-dimensional space of $F$-invariant linear forms, namely the dual space of $\IF\_{p^m} / im(F\_{|\IF\_{p^m}}-1)$. That is one-dimensional because $\dim im(F-1) = m - \dim ker(F-1) = m-1$. The problem of course is that there is no canonical one and these linear forms do not extend to $F$-invariant linear forms on field extensions of $\IF\_{p^m}$ in general (they extend iff the degree of the extension is not divisible by $p$).


Together with $\tau$ you can build an $F$-invariant linear form on $\IF\_{p^a} \otimes \bigcup\_{(p,m)=1} \IF\_{p^m}$ but that is as far as you can go.


