# If two monic polynomials of $\mathbb{Z}_p[X]$ (p-adic integer coefficient) are relatively prime modulo p, then do they generate $\mathbb{Z}_p[X]$?

I'm currently reading a paper of Rene Schoof, and I got stuck in a line. And I'm trying to check the above sentence. Although that seems to be elementary, I hope someone can give me a counterexample or a proof.

Thank you.

• Polynomial Bezout identity? – Seva Aug 1 at 4:24
• Your $\mathbf Z_p$ is presumably notation for the integers mod $p$, not the $p$-adic integers. In a PID, if two elements are relatively prime do you know if they generate the ring? – KConrad Aug 1 at 4:43
• I'm Sorry for the notation, by $\mathbb{Z}_p$ I mean the ring of p-adic integers. By modulo p, I mean modulo $p\mathbb{Z}_p[X]$. – gualterio Aug 1 at 4:45

If the two polynomials are $$f,g$$, then the $$\mathbb{Z}_{p}$$-module $$M := \mathbb{Z}_{p}[X]/(f,g)$$ is finitely generated (since at least one of $$f,g$$ is monic) and satisfies $$M \otimes_{\mathbb{Z}_{p}} \mathbb{Z}_{p}/(p) = 0$$, hence $$M = 0$$ by Nakayama.