For $n\geqslant 1$ define $$A_n=\mathbb{C}[X_0,X_1,\dots,X_n]\Bigg/\left(\sum_{i=0}^{n}X^2_i-1\right).$$
I would like to prove that $A_3$ is a unique factorization domain.
For $A_2$ it is not true because we can show that the factorization is not always unique. In general, $A_n$ is a Noetherian integral domain, so it suffices to prove that every irreducible element of $A_3$ is also a prime. There is an additional lemma which might be helpful: If $A$ is a Noetherian domain and $\pi$ its prime element, therefore $A[X]/(\pi X-1)$ being UFD implies that $A$ is UFD.
Any directions would be helpful. References?
