TITLE: Irreducibility of polynomials in two variables
QUESTION [74 upvotes]: Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in two variables, Acta Arith. 1997).
Does anyone know of similar results in the same vein? How about criteria over fields other than the complex numbers?

REPLY [21 votes]: Quite a useful result can be found in Schmidt's lecture notes on Equations over finite fields (Theorem III.1B in SLNM 536). Suppose that $K$ is any field and let $f(x,y)=c_0y^d+c_1(x)y^{d-1}+
\cdots+c_d(x) \in K[x,y]$, with $c_0 \neq 0$.  Let 
$$
\psi(f)=\sup_{1\leq
i\leq d}\frac{\deg c_i}{i}.
$$
Then $f$ is absolutely irreducible over $K$ provided that that $\psi(f)=m/d$ with $\gcd(m,d)=1$.
This shows, for example, that the polynomial $f(x,y)=g(x)-h(y)$ is irreducible when  $\deg g$ and $\deg h$ are coprime.