We consider a dynamical system consisting of a pair of commuting power series under composition, one noninvertible and another nontorsion invertible, of height one with coefficients in the $p$-adic integers. Assuming that each point of the dynamical system generates a Galois extension over the base field, we show that these extensions are in fact abelian, and, using results from the theory of the field of norms, we also show that the dynamical system must include a torsion series. From an earlier...

Let $K$ be a finite extension of ${\mathbb{Q}}_p$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma$ and ramification index $p^n$. We give a criterion in terms of the ramification numbers $t_i$ for a fractional ideal ${𝔓}^h$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄(K\Gamma ;{𝔓}^h)$. In particular, if $t_n-[t_n/p]\<p^{n-1}e$ then the inverse different can be free over its associated order only when $t_i\equiv -1$ (mod $p^n$) for all $i$. We give three consequences of this. Firstly, if $𝔄(K\Gamma ;S)$ is a Hopf order and $S$ is $𝔄(K\Gamma ;S)$-Galois...