(Ir)reducibility criteria for homogeneous polynomials

Solution 1:

In exactly $3$ variables, suppose a given homogeneous polynomial $F$ is reducible and suppose that the base field is algebraically closed. Then its factors describe projective curves which intersect by Bezout's theorem, and you can determine these intersection points by looking at the points in $\mathbb{P}^2$ where the partial derivatives of $F$ simultaneously vanish. If there are no such points, $F$ must be irreducible.

For example, $X^n + Y^n + Z^n$ must be irreducible by this criterion because the partial derivatives only simultaneously vanish at $(0 : 0 : 0)$.