Proving a Certain $\mathbb{C}$-Algebra is a Domain Using a Specified Method

Question

I would like to prove that $A = \mathbb{C}[x,y,z]/(x^2+z^2-1, x^2-y^2-z^2)$ is an integral domain.

I feel that it would be easy enough to prove the denominator is prime using the techniques of Grobner bases, but my professor suggests another way. He says to consider the inclusions $\mathbb{C}[x]$ $\subset$ $B = \mathbb{C}[x,z]/(z^2-(1-x^2))$ $\subset$ $A$ and use a result which says that $k[x,y]/(y^2-f(x))$ is an integral domain if and only if $f(x)$ is not a square in $k[x]$. This shows $B$ is a domain, but here I get stuck. I consider $A=B[y]/(y^2 - (x^2-z^2))$ and I can see that the denominator is irreducible, but $B$ is not a UFD, so irreducible is not necessarily equivalent to prime. How should I proceed?

Answer 1

Since the question contains a mistake, let me fix it.

The ring $R=\mathbb{C}[X,Y]/(X^2+Y^2-1)$ is a UFD because it is isomorphic to $\mathbb{C}[U,V]/(UV-1)$ (via the substitutions $U\mapsto X+iY$ and $V\mapsto X-iY$) and the last one is a ring of fractions of $\mathbb{C}[U]$ (with respect to the multiplicative system $\{1,U,U^2,\dots\}$).

Remark. If $R$ is an integral domain and $\alpha\in R$, then $R[X]/(X^2−\alpha)$ is an integral domain if and only if there are no non-zero elements $a,b\in R$ such that $b^2=\alpha a^2$.