Do any two coprime factors of $x^n-1$ over the $p$-adic integers $\mathbb{Z}_p$ which remain coprime over $\mathbb{F}_p$ generate comaximal ideals? Let $f,g$ be distinct irreducible factors of $x^n-1$ over $\mathbb{Z}_p[x]$ (polynomials over $p$-adic integers). Suppose $\overline{f},\overline{g}$ are coprime in $\mathbb{F}_p[x]$ - thus, the ideal generated by them $(\overline{f},\overline{g}) = 1$ in $\mathbb{F}_p[x]$. Must $(f,g) = 1$ in $\mathbb{Z}_p[x]$?
Note that $f,g$ are certainly coprime, but $\mathbb{Z}_p[x]$, coprime doesn't mean comaximal (e.g. $p,x$ are coprime but not comaximal).

Suppose $(f,g)\ne 1$, then they are contained in some maximal ideal $m\supset (f,g)$, but the maximal ideals of $\mathbb{Z}_p[x]$ are precisely the ideals of the form $(p,h(x))$, where $h(x)$ is irreducible and remains irreducible mod $p$. Thus, $\mathbb{Z}_p[x]/m\cong \mathbb{F}_p[x]/(\overline{h})$. This implies that $(\overline{h})\supset(\overline{f},\overline{g})$, but since $\overline{f},\overline{g}$ are comaximal, they generate the unit ideal, and so $\overline{h}$ must be a unit, contradicting the fact that $h$ is irreducible mod $p$.
This implies that $(f,g) = 1$.
