How to prove a polynomial in $\mathbb{Z}[x,y]$ is irreducible

Question

I had the following question:

Let $f(x,y) = y^5+xy^2+x \in \mathbb{Z}[x,y]$ and let:

$I_2 = (f(x,y),x-1,2)$ and $I_3 = (f(x,y),x-1,3)$ be two ideals in $\mathbb{Z}[x,y]$.

Prove that $f(x,y)$ is irreducible, and determine which of these ideals is maximal.

My ideas:

1. I am guesssing for the first part apply Eisenstein with $x$ as $(x)$ is prime ideal in $\mathbb{Z}[x,y]$? Is this correct?

2. For part $2$, I know that an ideal $I$ of $R$ is maximal iff $R/I$ is a field. How would I apply this here?

Thanks for any help.

Answer 1

For part 1, it's perfectly correc.

Hint for part 2: $\;\mathbf Z[x,y]/I_2\simeq (\mathbf Z/2\mathbf Z)[y]\bigm/\bigl(f(1,y)\bigr)= (\mathbf Z/2\mathbf Z)[y]\bigm/(y^5+y^2+1))$.

Obviously it has no root in $\mathbf Z/2\mathbf Z$, so you only have to determine whether it has a quadratic factor or not.

Similar method for $I_3$, but you'll have to compute in $\mathbf Z/3\mathbf Z$.