I'm trying to prove that $\overline{x}\in \frac{k[x,y,z]}{(xz-y^2)}$ is irreducible. (part of a problem 4.5 in Hulek's elementary algebraic geometry)
I attacked this by using the fact that every elt in $\frac{k[x,y,z]}{(xz-y^2)}$ has a unique representation of the form $f(x,z)y+g(x,z)$ and some elementary methods like comparing degree, resulting in a lot of pointless equations.
I want to complete the proof in this way, not using a kind of singularity or something hard(because TA said this method is valid), but I don't know how to.