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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 $\mathbb{R}^\times$?

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    $\begingroup$ If I have time tomorrow, I can elaborate on my comment from the other question about considering such a group as an algebraic group (which is probably one of the best view points for an arbitrary field). The basic idea is that this group can be given a scheme structure as spec of $F[t,t^{-1}]$. $\endgroup$ – Tobias Kildetoft Sep 11 '16 at 19:27
  • $\begingroup$ @TobiasKildetoft: Thank you. I will look forward to that. $\endgroup$ – John Doe Sep 11 '16 at 19:28
  • $\begingroup$ $F^{\times}$ can be a complicated group in general, and the assumption that it's the multiplicative group of a field isn't super helpful other than knowing that it's commutative. $\mathbb{R}^{\times}$ is just $\{ \pm 1 \} \times \mathbb{R}$, so it's about as hard to understand as $\mathbb{R}$. $\endgroup$ – Qiaochu Yuan Sep 11 '16 at 20:15
  • $\begingroup$ @TobiasKildetoft: I offered a bounty on the question. $\endgroup$ – John Doe Nov 14 '16 at 15:11
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    $\begingroup$ @Alex: the standard convention in English is that fields are commutative. Noncommutative or skew fields are usually called division rings. I think the convention might be different in French. $\endgroup$ – Qiaochu Yuan Nov 14 '16 at 16:00

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