$Y^3+XY^2+X^3Y+X$ is irreducible in $\mathbb{C}[X,Y]$

Question

I want to prove that the polynomial $$Y^3+XY^2+X^3Y+X$$ is irreducible in $\mathbb{C}[X,Y]$.
If I try to write it as product of two polynomials $g$ and $h$, I can see that in order for $Y^3$ to appear, $g$ must contain a term $\alpha Y^2$ and $h$ a term $1/\alpha Y$ or vice versa and similarly one must contain $\beta X$ and the other $1/\beta$ etc., but the cases seem too complicated.
Also, because the coefficient field is $\mathbb{C}$, I don't see how usual tricks like Eisenstein's can work.

Answer 1

Just consider your polynomial in $(\Bbb C[X])[Y]$ . $\Bbb C[X]$ is a PID, in particular a UFD. Now $X$ is an irreducible in $\Bbb C[X]$ . Then apply Eisentein's Criterion on $Y^3+XY^2+X^3Y+X$ . Since, $X|X,X|X^3$ and $X$ doesn't divide $Y^3$ nor does $X^2$ divide $X$ , we are done!

Answer 2

Assume your polynomial $P$ can be written as $g(X,Y)h(X,Y)$.

Then $g(0,Y)h(0,Y)=Y^3$, so either $Y|g(0,Y)$ and $Y|h(0,Y)$, or (say) $g(0,Y)=1$.

In the first case, $g(X,0),h(X,0) \in X\mathbb{C}[X,Y]$ and thus $X^2|X=P(X,0)=g(X,0)h(X,0)$ and we get a contradiction.

In the second case, $h(0,Y)=Y^3$, so $P(X,Y)=g(X,Y)h(X,Y) \in g(X,Y)(Y^3+X\mathbb{C}[X,Y])$. It follows $g=1$.