This is an exercice of an Algebraic Geometry course.
In a way to study all the irreducible varieties in the affine two dimensional complex space, we are looking for the prime ideals of $A=\Bbb C[X,Y]$. The exercice proposes to study it in different cases. We will also use localization of rings. Here is one case I'm struggling with.
So let $P$ be a prime ideal of $A$ such that $P \cap\Bbb C[X] = 0$ and $S = \Bbb C [X] \setminus 0 $. We see that $AS^{-1} \cong \Bbb C(X)[Y]$ and that $PS^{-1}$ is a prime ideal of $AS^{-1}$.
My problem is the next step where we have to show that if $P \neq 0$, then there exists a irreducible polynomial $f \in \Bbb C(X)[Y]$ such that $PS^{-1} = (f)$.
Can anyone help ?