Let $w$ be an algebraic element over $\mathbb{C}[x]$, with minimal polynomial $f(t)=c_mt^m+\cdots+c_1t+c_0$, $c_i \in \mathbb{C}[x]$.
Is it possible to find all prime elements of $\mathbb{C}[x]$ which remain prime in $R=\mathbb{C}[x][w]$? (in case $R$ has prime elements). Perhaps a partial answer should be in terms of the $c_j$'s.
Remarks: An irreducible element of $\mathbb{C}[x]$ is of the form $ax+b$, $0 \neq a,b \in \mathbb{C}$, because $\mathbb{C}$ is algebraically closed (we can replace $\mathbb{C}$ by any algebraically closed field). Also, $\mathbb{C}[x]$ is a UFD, so irreducibles= primes.
Non-example: $R= \mathbb{C}[x][x^{3/2}]$, $f(t)=t^2-x^3$ is the minimal polynomial of $w=x^{3/2}$. Notice that $x$ is prime in $\mathbb{C}[x]$, but it is not prime in $R$, because $xxx=x^{3/2}x^{3/2}$, namely, $x$ divides the product $x^{3/2}x^{3/2}$, but $x$ does not divide $x^{3/2}$. Actually, $R$ does not have prime elements, see this question.
This is a relevant question.