Let $D$ be a Noetherian UFD over a field $k$ of characteristic zero.
Let $w$ be an algebraic, non-integral element over $D$, and denote its minimal polynomial over $D$ by $h(T)=d_nT^n+\cdots+d_1T+d_0$, $d_j \in D$.
Denote $R=D[w]$; it is a Noetherian integral domain, but it is not necessarily a UFD.
When is every prime element of $D$ irreducible in $R$? (By 'when' I mean what should be the $d_j$'s, or additional assumptions on $D$).
Remarks: (1) $D$ is a UFD, so primes=irreducibles.
(2) $R$ is not necessarily a UFD, so it may not have prime elements, but it is Noetherian so every element of $R$ is a product (not necessarily unique) of irreducible elements of $R$.
(3) If $d \in D$ is prime/irreducible, it may happen that $d$ is reducible in $R$; for example: If $d_0$ is prime, then $d_0=-(d_nw^{n-1}+\cdots+d_1)w$ is reducible.
(4) Please notice that this question is almost identical to the current question; however, its answer actually answers a previous version of that question, so that question is unanswered. Please also see this unanswered question.
Thank you very much for any hints and comments!