Is $\mathbb{C}[x,y]/(x^2+y)$ a UFD?

Question

I guess if we find two factorisations which are the same then we could say that it is not a UFD. I saw the related question Is $\Bbb{R}[X,Y]/(X^2+Y^2)$ a UFD or Noetherian? and I tried following the same example.

The thing I don't understand is why we can reduce it to $x^2+y=0$.

And could somebody give me a hint regarding the factorisation. I couldn't find any since the fact that y has degree 1 makes it hard but I am not sure at all.

Thanks in advance

Answer 1

First, let's note that $\mathbb{C}[x, y] / (x^2 + y) \cong \mathbb{C}[x]$ under the mapping sending $x$ to $x$ and $y$ to $-x^2$.

Since $\mathbb{C}$ is a field, $\mathbb{C}[x]$ is a Euclidean domain, hence a UFD.

And of course being a UFD is a property which is invariant under ring isomorphism.