Is $\mathbb{C}[x,y]/(x^3+y^3−1)$ is a UFD or not?

Question

I'm wondering if $\mathbb{C}[x,y]/(x^3+y^3−1)$ is a UFD or not. I know that a Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal, and I know that the krull dimension of $\mathbb{C}[x,y]/(x^3+y^3−1)$ is 1, and every nonzero prime ideal is of the form $(x-a,y-b)$, where $a^3+b^3=1$. But I don't know whether such a prime ideal is principal or not. Or maybe there are other methods to solve this problem.

Could anyone give me a hand please? Thanks in advance.

That looks like a Dedekind domain whose fraction field is that of an elliptic curve. Being an elliptic curve would surely block it from being a UFD. — Angina Seng, Jun 23 '20 at 09:34

score 1 · Answer 1 · answered Jun 23 '20 at 09:14

1

Hint. Let $\alpha$ the class of $x$ and let $\beta$ the class of $y$.

Show that $\alpha$ is irreducible, but that $(\alpha)$ is not a prime ideal, for example. For the last part, notice that $\alpha^3=(1-\beta)(1+\beta+\beta^2)$.

answered Jun 23 '20 at 09:14

GreginGre

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Is $\mathbb{C}[x,y]/(x^3+y^3−1)$ is a UFD or not?

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