Suppose $K/\mathbb{Q}_p$ is a finite extension with residue field $k$. Fix a uniformizer $\pi\in K$ and choose a coherent sequence $(\pi^{1/p^n})$ of $p$-power roots of $\pi$, and let $K_\infty/K$ be the extension of $K$ generated by these roots. The field $K_\infty$ is arithmetically pro-finite and so we can consider its field of norms, which happens to be isomorphic to the function field $k((u))$. Let $\mathcal{R}$ be a Cohen ring for $k((u))$, equipped with a lift $\varphi$ of Frobenius. Conventionally, we take this to be the $p$-adic completion of $W(k)[[u]]_{(u)}$ and $\varphi$ sends $u$ to $u^p$.
Then we have equivalences: $G_{K_\infty}$-reps $\Leftrightarrow$ $G_{k((u))}$-reps $\Leftrightarrow$ etale $\varphi$-modules over $\mathcal{R}$.
Although I've known this basic theory for a while, I realized I did not know the answers to some elementary questions, and couldn't immediately find them in the literature. It seems to me that these are toy examples of the theory and should appear somewhere!
Question 1: Let $\chi$ be the restriction of the $p$-adic cyclotomic character to $G_{K_\infty}$. Is there a nice description of the corresponding character of $G_{k((u))}$? In other words, what is the norm field extension of $k((u))$ corresponding to $K_\infty(\mu_{p^\infty})$?
Question 2: By the second equivalence, there should be a 'cyclotomic' period in $\mathcal{R}$ that transforms under $G_{k((u))}$ by the character that is the answer to Question 1. Can we write it down explicitly?
