let $\mathbb{Z}_p$ be the ring $p$-adic integers where $p$ is an arbitrary prim. Is there way to classify all prime ideals in $\mathbb{Z}_p[X]$ in a meaningful way. That is, consists the set of primes of exactly following four types?
-zero ideal
-$(p)$
-$(f)$; $f \in \mathbb{Z}_p[X]$ irreducible
-$(f,p)$; $f$ irred
Are there no other? can it be shown with eg Hensel's lemma?
It looks like you found all of the primes.
The ring $\mathbb Z_p$ of $p$-adic integers is a Noetherian, local, integral domain of dimension $1$. The problem of determining the spectrum of a polynomial ring $R[x]$ when the coefficient ring $R$ is a countable, Noetherian, semilocal, integral domain of dimension $1$ was solved by Heinzer and Wiegand in
W. Heinzer and S. Wiegand, Prime ideals in two-dimensional polynomial rings. Proc. Amer. Math. Soc. 107 (1989), no. 3, 577-586.
(Semilocal = finitely many maximal ideals.)
The case where the coefficent ring is uncountable was treated in
C. Shah, Affine and projective lines over one-dimensional semilocal domains. Proc. Amer. Math. Soc. 124 (1996), no. 3, 697-705.
C. Shah, One-dimensional semilocal rings with residue domains of prescribed cardinalities. Comm. Algebra 25 (1997), no. 5, 1641-1654.
Shah's work contains some mistakes in cardinal arithmetic, which were discovered by Greg Oman. Her work is correct in its description of the types of primes in $R[x]$, but not the number of primes of each type. Shah died in 2005 before Oman could notify her, so we showed how to correct the mistakes in:
Keith A. Kearnes and Greg Oman, Cardinalities of residue fields of Noetherian integral domains. Comm. Algebra 38 (2010), no. 8, 3580-3588.