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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.

Vinay Deshpande
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    If the polynomial is irreducible, then clearly primality follows. For irreducibility, consider it as a polynomial just in $y$, and observe the form the factors must have. – Sarvesh Ravichandran Iyer Aug 10 '20 at 05:28

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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 $g(x)$.

Angina Seng
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