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 characterize (in terms of the $c_j$'s) all algebraic elements $w$, such that $R=\mathbb{C}[x][w]$ has no prime elements?

Please see this related question, in which the special case $k[x^2][x^3]$ is dealt with (notice that in this special case, $m=2$ with $c_2=1$, namely, $w$ is integral).