Let $F$ be a field, and $F[x,y]$ be the ring of polynomials in two variables and we know that $F[x,y]$ is integral domain but not Principal Ideal Domain. We know that $y^2-x$ is irreducible in $F[x,y]$.
How to prove that $(y^2-x)$ is a prime ideal in $F[x,y]$?
If we let $f$ and $g$ be in $F[x,y]$, such that $y^2-x\mid fg$, can we claim that $y^2-x\mid f$ or $y^2-x\mid g$?