I was wondering if anyone could think of an elegant proof that the polynomial

$\displaystyle f(x,y,z)=z^2-xy$

is irreducible in $\displaystyle k[x,y,z]$.

(My problem had $\displaystyle \mathrm{char}k\neq 2$, but I don't think it's relevant for this part.)

I have one proof, but it's basically brute force, using that if $\displaystyle f=gh$ then (WLOG) $\displaystyle \mathrm{deg}_x(g)=0$, and just writing out what the factors look like. I was wondering if there was a... nicer way.