# Factoring two-variable polynomial over algebraic closure of $\mathbb{F}_p$.

I am currently working on algebraic projective curves and I feel I'm missing some concrete calculations.

I would therefore like to do a few computations on my own. For that I need first a way of actually creating an irreducible curve.

I am wondering therefore if there is an algorithm to factor a two variable polynomial P(X,Y) in the algebraic closure of a given finite field $\mathbb{F}_p$. Intuitively I think there might be a way to brute-force this: Possibly create a finite field of high enough cardinal and then just try every possible factor(since there is a finite possible number of them.) That however seems to me extremely ineffective, so I was wondering if there is a better way of doing it. Some sort of multi-variable Berlekamp algorithm.

• It's easier to just write down irreducible polynomials, and there are many polynomials it's easy to prove are irreducible. For starters, try showing that $y^2 - x^n$ is irreducible iff $n$ is odd. – Qiaochu Yuan Nov 28 '17 at 19:10
• Thank you, I will definitely work on these in the meantime. I still would like to know if there is some sort of general algorithm though. – Keen Nov 28 '17 at 19:18