$\mathbb{C}[x,y,z]/(x^2+y^2+z^2-1)$ is not a UFD

abstract-algebraalgebraic-geometrycommutative-algebrapolynomial-ringsunique-factorization-domains

Wiki says that the coordinate ring $$\mathbb{C}[x,y,z]/(x^2+y^2+z^2−1)$$ of the complex sphere is not a unique factorization domain. I want to know why it is not a UFD.

We denote $$X,Y,Z$$ the residue class of $$x,y,z$$. Obviously, we have $$(X+iY)(X-iY)=(1+Z)(1-Z)$$ in $$\mathbb{C}[x,y,z]/(x^2+y^2+z^2−1)$$. But how to prove the irreducibility? Maybe there are some deep techniques of commutative algebra or algebraic geometry should be used in this question.

Theorem. Suppose $$A$$ is a noetherian domain. Then $$A$$ is a UFD iff $$X=\operatorname{Spec} A$$ is normal and $$\operatorname{Cl} X=0$$.
It's clear that with $$A=\Bbb C[x,y,z]/(x^2+y^2+z^2-1)$$, $$X$$ is normal (it's even smooth), so we can compute $$\operatorname{Cl} X$$ to see whether $$A$$ is a UFD or not.
Let $$X'=V(-x_0^2+x_1^2+x_2^2+x_3^2)\subset\Bbb P^3$$. Then $$A$$ is the coordinate algebra of $$D(x_0)\cap X'$$, and the class groups of $$X$$ and $$X'$$ are related by the exact sequence $$\Bbb Z\to \operatorname{Cl} X'\to\operatorname{Cl} X\to 0,$$ where the first map sends $$1$$ to the class of $$V(x_0)\cap X$$. As the middle group is $$\Bbb Z^2$$ from identifying $$X'$$ with $$\Bbb P^1\times\Bbb P^1$$, $$\operatorname{Cl} X\neq 0$$ and thus $$A$$ is not a UFD. (NB: if you're reading this and wondering how this all breaks for the real case mentioned at wikipedia, the class group of $$X'$$ depends on the base field!)