Is $X^2Y+Y^2X^{2018}+X+Y+1 \in \mathbb F_2[X,Y]$ irreducible?

Question

This is my problem:

Is the polynomial $X^2Y+Y^2X^{2018}+X+Y+1 \in \mathbb F_2[X,Y]$ irreducible?

I only have one theorem I can use to show that a polynomial is reducible, but I've already seen myself that it doesn't hold in this case, so my assumption right now is that the above polynomial is irreducible, but I don't know how to show that.

I know of the following ways to do that: Eisenstein, reduction, root criterion. I don't think I can use root criterion here and I'm pretty sure I can't use Eisenstein here either (or can I?). So I tried using reduction, but no matter what prime element I use, it won't work. Can someone help me out?

Thanks in advance.

Answer 1

This is $$(X^{2018})Y^2+(1+X)^2Y+(1+X).$$ Think of this as an element of $R[Y]$ where $R=\Bbb F_2[X]$, which is a PID. Now $X+1$ is an irreducible in this PID, so we can apply Eisenstein.