Prove that $X^3+Y^3+X^2Y+XY^2+Y^2-Y$ is irreducible in $\mathbb{Z}[X,Y]$

Question

I want to prove that $g=X^3+Y^3+X^2Y+XY^2+Y^2-Y$ is irreducible in $\mathbb{Z}[X,Y]$.

Therefore I wrote $g$ as polynomial in $X$ with coefficients in $\mathbb{Z}[Y]$ like $$g=X^3+(Y)X^2+(Y^2)X+(Y^3+Y^2-Y).$$ Is this a primitive polynomial? And how do I use Eisenstein's criterion to show that this is irreducible?

Answer 1

Any value of $Y$ that you plug in that gives you a polynomial in $X$ that satisfies Eisenstein holds is good enough for your purposes.

Answer 2

Suppose $g(X,Y)=f(X,Y)h(X,Y)$. Fix a prime $p\in\mathbb{Z}$. Then, $g(X,p)=f(X,p)h(X,p)$, so either $f(X,p)=1$ or $h(X,p)=1$ by Eisenstein's criterion. Since we can do this for any prime, either $f(X,p)=1$ for infinitely many primes or $h(X,p)=1$ for infinitely many primes. Say this holds for $f$. Then $f(X,Y)-1=0$ has infinitely many solutions, so $f(X,Y)=1$.

Answer 3

I suggest you to use this form of the Eisenstein's criterion. Then the integral domain is $D=\mathbb Z[Y]$, and the prime ideal can be chosen $\mathfrak p=(Y)$. The only thing to show is $Y^3+Y^2-Y\notin (Y^2)$, but this is pretty clear. Now notice that $g$ is primitive, because its leading coefficient is $1$.