Iterated extensions and the ramification dichotomy

Abstract

Let $K/{\mathbb{Q}}_p$ be finite and let $f\in {\mathcal{O}}_K[X]$ be monic, of degree at least two, with $f^{\prime }(X)\in {\mathfrak{m}}_K{\mathcal{O}}_K[X]$, equivalently $\bar{f}\in k[X^p]$. For a compatible inverse branch $f(t_{n+1})=t_n$ with $t_0\in {\mathcal{O}}_K$, put $K_n=K(t_n)$ and $K_{\infty }={\bigcup }_n{K}_n$. We prove that $K_{\infty }/K$ is either unramified or deeply ramified. More precisely, once ramification appears, the ramification indices over the maximal unramified subfields tend to infinity and the finite-level differents are unbounded. In the Frobenius-type case $f(X)\equiv X^{p^a}(\mathrm{mod}{\mathfrak{m}}_K)$ the unramified alternative is trivial, so $K_{\infty }=K$ or $K_{\infty }/K$ is deeply ramified. After completion, the non-unramified alternative gives perfectoid fields and examples show that APF property need not hold at the algebraic level.