whether $(y^2-x^3-x^2)$ is a prime ideal of $\mathbb{C}[x, y]$?
Many useful results for one variable (for eg polynomial ring over a field is Euclidean domain) fail in case of multiple variables.
Can't guess factors of $y^2-x^3-x^2$ in $ \mathbb{C}[x, y]$.
Another way is to define a homomorphism to an integral domain with kernel $(y^2-x^3-x^2)$ but can't guess the integral domain.
Please give a hint. Similar posts (if any) from this site are also welcome.
Consider a polynomial of the form $f(x,y)=y^2-g(x)$ over $\Bbb C$. I claim that $f$ is reducible over $\Bbb C[x,y]$ iff $g(x)$ is a square in $\Bbb C[x]$ (which $x^3+x^2$ isn't).
As $f$ has degree $2$ in $y$, and is monic in $y$, a non-trivial factorisation of $f(x,y)$ must take the form $(y+a(x))(y+b(x))$. This is only possible if $a(x)=-b(x)$ and both are square roots of $g(x)$.