I have a sort of follow-up question to Representation theory of $\mathbb{R}$?
If we have a finite field $F$, then $F^{\times}$ is a cyclic group. And so I believe that I understand how the representations (homomorphisms $G \to GL(V)$) work out.
My question is how this works when $F$ is an infinite field?
I get from the other answer, that considering representations of fields (considered as groups) is a bit tricky, but I am wondering what can be said when the one takes the units.
If this is too complicated or too broad, then I would welcome a reference.
For example, what is the representation theory of $\mathbb{R}^\times$?