Let $D$ be a (Noetherian) UFD, $T$ transcendental over $D$, $p=p(T) \in D[T]$ irreducible (=prime, since $D[T]$ is also a UFD).
My question: Given an irreducible element $d \in D$, is it possible to tell when $d$ remains irreducible as an element of $E:= D[T]/(p)$? Namely, is there a special form of $p$ which guarantees that $d$ remains irreducible in $E$?
Write $p=p_nT^n+\cdots+p_1T+p_0$, where $p_n,\ldots,p_1,p_0 \in D$. In the special case where $d=p_0$ (and $p_0$ is irreducible in $D$), it is clear that $p_0$ is reducible in $E$: $p_0 = -T(p_nT^{n-1}+\cdots+p_1)$. (For example: $D=\mathbb{Z}$, $p=T^2+3$, then $E \cong \mathbb{Z}[\sqrt{-3}]$. $d=3$ is irreducible in $\mathbb{Z}$, but reducible in $E$: $3 = -\sqrt{-3} \sqrt{-3}$).
But what if $d \neq p_0$?
Remarks/ideas:
(1) If $p$ is of degree $1$, then $E \cong D[w]$, $w \in Q(D)$ ($Q(D)$ denotes the field of fractions of $D$). Can we tell if $d$ remains irreducible in $E$? Actually, I prefer to assume that $p$ has degree $\geq 2$ (or $\geq 3$), but I guess it is better/easier to start with considering low degrees $1,2$.
(2) What if $p$ is monic? Does this help?
See also this recent question.