TITLE: Irreducibility of Polynomials in $k[x,y]$
QUESTION [22 upvotes]: I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible.  
For example, in problem I.5.1(b) we are studying the polynomial $f=xy-x^6-y^6$. How can I show this is irreducible?  
The standard tricks (Eisenstein, linearity in one variable, etc...) don't seem to work here.
Edit: Some of the other polynomials in the problem are also giving me trouble. They are $g=x^3-y^2-x^4-y^4$ and $h=x^2y+xy^2-x^4-y^4$. Any ideas for these?

REPLY [7 votes]: There is an easy way to see that all of $f,g,h$ are irreducible using Newton polytopes. The method is described in this math overflow answer.
For example, in the case of $h=x^2y+xy^2-y^4-x^4$ the polynomial has nonzero coefficients at $(2,1)$, $(1,2)$, $(0,4)$ and $(4,0)$. Going clockwise around convex hull of this, we move by vectors
$$(-1,1), (-1,2), (4,-4), (-2,1).$$
This sequence cannot be partitioned into two such that each of them has sum zero, therefore $h$ is irreducible.