Let $A \subseteq A[w] \subseteq C$ be three Noetherian integral domains, over a field $k$ of characteristic zero, with $A \subseteq C$ an algebraic ring extension (in particular, $w$ algebraic over $A$).

Further assume that: **(1)** $A$ and $C$ are unique factorization domains (UFD's).

**(2)** $A \subseteq C$ is root closed, namely, for all $n$, if $c \in C$ satisfies $c^n \in A$, then $c \in A$.
   
**(3)** $w$ is an irreducible element of $C$ (hence it is also an irreducible element of $A[w]$). Notice that, since $C$ is assumed to be a UFD, the assumption that $w$ is irreducible in $C$ implies that it is prime in $C$
(but it can be non-prime in $A[w]$).
 
**(4)** $w$ is a prime element of $A[w]$.   

> Is it true that $A[w]$ must be a UFD? Or, is there a counter-example?

**Remark:** Without assuming (2),(3),(4) it is easy to find a counter-example:
$k[t^2] \subset k[t^2][t^3] \subset k[t]$ ($w=t^3$). Clearly, $k[t^2]$ and $k[t]$ are UFD's, but $k[t^2][t^3]$ is not a UFD, since $t^3$ is an irreducible element of $k[t^2][t^3]$ which is not prime: $t^3t^3=t^2t^2t^2$.

Somewhat relevant questions are: [1][1], [2][2].


  [1]: https://mathoverflow.net/questions/269516/characterizing-all-simple-algebraic-ring-extensions-of-mathbbcx-having-no
  [2]: https://math.stackexchange.com/questions/2276356/a-simple-algebraic-ring-extension-of-a-ufd-having-no-prime-elements