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 $f'(X)\in\mathfrak m_K\mathcal O_K[X]$, 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 $K_\infty=\bigcup_nK_n$. 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 $f(X)\equiv X^{p^a}\pmod{\mathfrak m_K}$ 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.