Let $R$ be a Dedekind domain with fraction field $K$, and let $\mathfrak p$ be a maximal ideal of $R$. Let $f\in R[x]$ be a monic, separable polynomial and let $N/K$ be a splitting field of $f$.
A well known theorem of Dedekind in the case $R=\mathbb Z$ states that if $\bar f(x)$ factors as $\bar g_1(x)\cdots \bar g_m(x)$, then the decomposition group of any prime of $N$ lying over $\mathfrak p$ can be generated by an element whose permutation action on the roots of $f(x)$ is a product of cycles of length $(\deg g_i)$ for every $i$.
My first question is whether there are any generalizations of this result to other Dedekind domains. I'm particularly interested in the case where $R$ is the polynomial ring $\mathbb Q[T]$. Of course Dedekind's result uses the fact that $\mathbb Z$ has finite quotients, which won't hold more generally, but is there anything that can be said in the general case, or in the case $R=\mathbb Q[T]$?
My second question concerns a complement to Dedekind's theorem due to Beckmann (Journal of Algebra 164, pp 415-429), also in the case $R=\mathbb Z$. Assuming that $f(x)=g_1(x)^{e_1}\cdots g_m(x)^{e_m}+p\cdot h(x)$, where $\bar h(x)$ is not divisible by any $\bar g_i(x)$ and $p$ does not divide any $e_i$, Beckmann shows that the inertia group of any prime of $N$ above $p$ is generated by an element whose permutation action on the roots is a product of $(\deg g_i)$ $e_i$-cycles for each $i$.
Can Beckmann's result be generalized to other Dedekind domains $R$? And is there in the literature a proof of this result based on standard theory of decomposition and inertia groups?
