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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?

anomaly
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    Won't the argument from $\Bbb{Z}[x]$ carry over to this case? With the obvious modifications that $\Bbb{Z}_p$ only has a single prime. – Jyrki Lahtonen May 09 '20 at 21:19
  • @JyrkiLahtonen that's definitely the right idea. $\operatorname{Spec}(\mathbb{Z}_p[X])$ is the product of the schemes $\operatorname{Spec}(\mathbb{Z}_p)$ and $\operatorname{Spec}(\mathbb{Z}[X])$, which tells you more explicitly how to handle the fact that the only non-zero prime in $\mathbb{Z}_p$ is $(p)$. – diracdeltafunk May 09 '20 at 21:24
  • well, up to now I not see any reason why the same strategy as for integers should fail here... –  May 09 '20 at 21:29
  • I think having just one prime ideal might be a nontrivial obstacle - I've seen it mentioned that this fact is true for Dedekind domains with infinitely many prime ideals but presumably they would not have excluded those with finitely many if it were always true. –  May 09 '20 at 22:51

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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.

Keith Kearnes
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