Factorization into irreducible factors I am trying to find a factorization of the following polynomial $f=X^{4}Y^{2}+X^{3}+X^{2}Y^{4}+XY^{2}$ into irreducible factors over $\mathbb{Z}_{2}[X,Y]$.
I know that $\mathbb{Z}_{2}[X,Y]$ can be written as $\mathbb{Z}_{2}[X][Y]$, so we have to bring first $f$ in the form of $\sum_{i=1}^{n}f_{i}Y^{i}$, where every $f_{i}$ is an irreducible polynomial of the ring $\mathbb{Z}_{2}[X]$, but i have problems with this and in general i don't understand how to show the factorization in cases of $\mathbb{Z}[X,Y]$, $\mathbb{Q}[X,Y]$ etc.
Can anybody help me or give me a hint how to find this factorization? Thank you in advance!

Hint: Group the terms for which the sums of exponents is the same, then factor the appropriate power of the form $X^aY^b$ within each group, then notice other obvious factors.
