Let $R=K[x,y,z]/(x^2-yz)$, where $K$ is a field. Show that $R$ is an integral domain, but not a unique factorization domain.
My idea: Since $K$ is a field, and I want to show $R$ is an integral domain.
For this it is sufficient to show that the ideal $I=(x^2-yz)$, is a prime ideal. For this I need to show the polynomial $(x^2-yz)$ is irreducible in $K[x,y,z]$.
Can anyone suggest me how I move for irreducibility here.
My attempt for irreducibility is:
suppose that the polynomial is reducible, then $x^2-yz=(x+a)(x+b)=x^2+(a+b)x+ab$.
This implies, $a+b=0$ and $ab=yz$. I didn't find any value of $a$ and $b$, which satisfies both the equations.
Can anyone suggest me some hint for irreducibility in multivariable polynomial.