Let $k$ be a field, $a\in k$, and for all $b\in k$, $a\neq b^2$. I want to check if the ring $k[x,y,z]/(x^2-ay^2)$ is an integral domain.
I need to show that $(x^2-ay^2)$ is a prime ideal. If $fg\in (x^2-ay^2)$, I don't know how to show $(x^2-ay^2)$ contains $f$ or $g$ ?
$k[x, y, z]$ is a unique factorization domain which helps: Now one can just show that $x^2-ay^2$ is an irreducible element.
You may approach this in various ways. One way that is quite convenient is to note that in $L[x,y,z]$ where $L=k(\sqrt{a}),$ one has $x^2-ay^2=(x-\sqrt{a}y)(x+\sqrt{a}y),$ and the two factors are irreducible. This implies that if $x^2-ay^2$ were to decompose in $k[x, y, z]$ as well, its irreducible factors would have to be $x-\sqrt{a}y, x+\sqrt{a}y$ at least up to $L$-scalar multiple. Now one concludes, since no nonzero $L$-scalar multiple of the polynomials have coefficients in $k$.
