irreduciblity in $\mathbb C[x,y,z]$

Question

Show that the polynomial $g(x,y,z)=z^{1000}+x+\sum_{i=1}^{999} x^iy^iz^i $ is irreducible in $\mathbb C[x,y,z]$.

I think to use Eisenstein's Criteria but I cant..? Somebody help me?

Answer 1

Ok, now it is an easy one:

Interpret $g$ as an polynomial in $z$ over the factorial ring $\mathbb C[x,y]$ and use Eisenstein with respect to the prime $x \in \mathbb C[x,y]$