Quotient $K[x,y]/(f)$, $f$ irreducible, which is not UFD

Question

Does anyone know an irreducible polynomial $f \in K[x,y]$ such that the quotient $K[x,y]/(f)$ is not a UFD? Is it known when this quotient is a UFD?

Thanks.

Answer 1

Yes, I think that the standard example is to take $f(x,y)=x^3-y^2$. Then when you look at it this is the same as the set of polynomials in $K[t]$ with no degree-one term. That is, things that look like $c_0+\sum_{i>1}c_it^i$, the sum being finite, of course. You prove this by mapping $K[x,y]$ to $K[t]$ by sending $x$ to $t^2$ and $y$ to $t^3$. I’ll leave it to you to check that the kernel is the ideal generated by $x^3-y^2$, that the image is what I said, and that this is not UFD.