# Math Help - Prove z^2-xy is irreducible.

1. ## Prove z^2-xy is irreducible.

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

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

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

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

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

2. Just a suggestion: you might be able to give some kind of geometric argument. If you decompose $f$, then it would have to be into linear terms, which are hyperplanes. You could try to claim that your surface cannot be the product of hyperplanes.