Consider a monic irreducible polynomial $P \in \Bbb Z[X]$ (equivalently, by Gauss lemma, it is irreducible in $\Bbb Q[X]$). What do the rings $$ R = \varprojlim_n \Bbb Z[X] / (P^n) \qquad\text{and}\qquad S = \varprojlim_n \Bbb Q[X] / (P^n) $$ look like? How are they related to the $p$-adic numbers (for some prime $p$)? How do they relate to $\Bbb Z[[X]]$ ? I first saw this construction here (hem!).
If $P=X$, then we have $R = \Bbb Z[[X]], S = \Bbb Q[[X]]$ (in particular, it is not true that $S = R \otimes_{\Bbb Z} \Bbb Q$), and $R/(X-p) \cong \Bbb Z_p$ for every prime $p$. But what do we get if for instance $P = X^2 + 1$ ? Do we get someting like $R = \Bbb Z[i][[X]]$ ?
Thank you!