Let $\mathbb{F}_p((x))$ be the local field of formal laurent series over the finite field $\mathbb{F}_p$ considered as a locally compact field.
What are the additive continuous characters $\mathbb{F}_p((x)) \to \mathbb{C}^{\times}$?
What are the multiplicative characters $\mathbb{F}_p((x))^{\times} \to \mathbb{C}^{\times}$?
I know how characters of $\mathbb{Q}_p$ are all built out of taking the sum of the fractional part of the $p$-adic expansion which is well defined upto an integer and I understand why it's continuous. For $\mathbb{F}_p((t))$ however I have no idea where to start. The presence of the finite field confuses me when I try to define a homomorphism.
I also know that $\mathbb{F}_p((x))^{\times}=\mathbb{F}_p[[x]]^{\times}\times x^\mathbb{Z}$. So it will presumably be enough to understand additive characters of $\mathbb{F}_p[[x]]$. Unfortunately here i'm stuck with the same problem as before...
This is a subject that I fear that I have not paid as much attention to as I should have. If any expert would like to chime in and accuse me of vapidity (or error) in what I say below, I will step back.
Let’s call our field $\Bbb F_p((x))=k$ and its ring of integers $\Bbb F_p[[t]]=\mathfrak o$. First, for the additive characters, note that, when we look modulo a basic neighborhood of $0$, i.e. $t^m\mathfrak o$, we have $$ X_m=k/t^m\mathfrak o\cong\bigoplus_{j<m}\Bbb F_p\,, $$ an infinite direct sum. This, of course, is radically different from the situation for $\Bbb Q_p$. I guess that (since everything goes into the $p$-th roots of unity in $\Bbb C^\times)$ the set of characters of $X_m$ is $Y_m=\prod_{j<m}C_p$. where by $C_p$ I just mean a cyclic group of order $p$. Now, since $k^+$ is the inverse limit of all the $X_m$, I guess what you want is the direct limit of the $Y_m$’s, but I have no feel for what this is. That’s all I can say about $k^+$.
For the multiplicative characters, the story is far more interesting. You have correctly identified $k^\times=x^{\Bbb Z}\oplus\mathfrak o^\times$, but $\mathfrak o^\times$ is equal, in turn, to $\Bbb F_p^\times\oplus(1+x\mathfrak o)$. What’s inside the parentheses here is, explicitly, $1+x\Bbb F_p[[x]]$, the principal units of $\mathfrak o$. And the structure of this has nothing whatever to do with any additive structure. Indeed, in this case the principal units are the direct product (not sum) of copies of $\Bbb Z_p$, each rank-one factor being the span of (as one possible choice among many) $1+x^n$, where the $n$ here run through all the integers prime to $p$. Explicitly, then, $$ 1+x\mathfrak o=\prod_{n:(n,p)=1}(1+x^n)^{\Bbb Z_p}\,. $$ The reason for this is that you can raise any series in $1+x\mathfrak o$ to any $p$-adic integer, ’cause the powers $(1+x)^{p^m}$ go to $1$ as $m\to\infty$. Now, for any one factor here, the continuous characters of $(1+x^n)^{\Bbb Z_p}$ into $\Bbb C^\times$ are the (continuous) characters of $\Bbb Z_p$ into $\Bbb C^\times$, and these are just the group of torsion elements of $\Bbb C^\times$ of $p$-power order. You may recognize this group as $\Bbb Q_p/\Bbb Z_p$. So it seems to me that the continuous characters of $1+x\mathfrak o$ into $\Bbb C^\times$ are $$ \bigoplus_{n:(n,p)=1}\Bbb Q_p/\Bbb Z_p\,. $$ I just hope I haven’t missed any subtleties, and I hope this will prove to be of some help.