Let $f \in \mathbb{C}[x,y,z]$ be a non-singular homogeneous polynomial. Then $f$ is irreducible.
This is a passage in Miranda "Algebraic Curves and Riemann Surfaces" book. He claims that it is a basic theorem, but I couldn't prove it. Any help? Thanks!
In the meantime I found a solution:
Suppose $f=A(x,y,z)B(x,y,z)$, both homogeneous (here I am using the fact that factors of homogeneous polynomials are homogeneous, which is easy to prove). By basic algebraic geometry, the two curves $\{A=0\}$ and $\{B=0\}$ intersect in some point $P$. Then by the rule for the derivative of the product it's easy to see that also $f(P)=0$, so $P$ is a singular point of $f$, contradiction.