Is $k[x,y,z]/(x^2+y^2-z^2)$ a UFD?

Let $k$ be an algebraically-closed field of characteristic not two. Then is the ring $$k[x,y,z]/(x^2+y^2-z^2)$$ a UFD? I admit that $k[x,y,z]/(xy-z^2)$ is not a UFD.

Solution 1:

The problem is more difficult than it first appears. Easy enough it is to say "Oh in the quotient ring we just have $z^2 = (x+iy)(x-iy)$ and so it is not a unique factorization domain". Though, one needs to prove that $x+iy, x-iy$ and $z$ are really irreducible elements in the quotient ring.

However you can reduce your problem to what you already know about $k[x,y,z]/(xy - z^2)$: I claim that in fact your ring is isomorphic to this! Indeed define variables \begin{eqnarray*} u&:=& x+iy\\ v &:=& x-iy .\end{eqnarray*}

and note that $k[x,y,z] = k[u,v,z]$ by using that $x = \frac{u+v}{2} $ and $ y = \frac{u-v}{2i}$. Then $$k[x,y,z]/(x^2 + y^2 - z^2) \cong k[u,v,z]/(uv - z^2)$$

where the ring on the right hand side you already know is not a UFD, so you have reduced your problem to what you already know!