Local fields, iterated extensions, and Julia Sets

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. For $\alpha \in K$, let $K_{\mathrm{\infty }}$ be the extension obtained by adjoining all iterated preimages of $\alpha$ under a unicritical polynomial $f_c(z)=z^{\mathrm{\ell }}-c\in K[z]$. We study the extension $K_{\mathrm{\infty }}\mathrm{/}K$ and show that its qualitative behavior depends only on the valuation of $c$. This removes the previous restrictions on $\mathrm{\ell }$ in work of Anderson–Hamblen–Poonen–Walton and completes the classification for all $\mathrm{\ell }\ge 2$. We also relate the ramification to the structure of the Berkovich Julia set of $f_c$.