Given a field of the form $\mathbb C(t)[g]$, where $t$ is transcendental over $\mathbb C$ and $g$ is algebraic over $\mathbb C(t)$ do I always find an irreducible polynomial $F\in \mathbb C[X,Y]$ such that $Quot(\mathbb C[X,Y]/(F))\cong \mathbb C(t)[g]$?
I thought about taking the minimal polynomial of $g$ over $\mathbb C(t)$ and killing the denominators, but I couldn't manage writing a serious proof.