# Irreducible Polynomials in two complex variables

I am seeking methods to see if a polynomial $f \in O(\mathbb{C^2},0)$ is irreducible. The subject is really new to my and I am studying for myself, for which I don't see about this subject.

Theorem: The intersection of an irreducible local divisor $D_f$ with another effective local divisor $D_g$ is isolated if and only if the germ $g\left( \tau \right)$ is not identically zero, and multiplicity $D_{f}\overset{0}{.}D_{g}$ of this intersection is equal to ther order $ord_{0}\left( g\left( \tau \right) \right)$ I need examples, therefore, need to read a little about the criteria for irreducible polynomials in two complex variables – Keinohrhasen Sep 14 '10 at 1:06
Here's one idea: if $f(x, y)$ is reducible, then the projective closure of $f(x, y) = 0$ has at least two components, which intersect by Bezout's theorem. These intersections are singularities and therefore can be found by setting all partial derivatives to zero. If no singularities exist then $f$ is irreducible.