# irreducible polynomials over $\mathbb{F}_p[x_1,x_2]$

Recently I was reading about a irreducibility test for polynomials over the ring $\mathbb{F}_q[x]$. It is layed on the fact that the product of all monic irreducible polynomials whose degree divides $k$ is equal to $x^{q^k}-x$ for every $k\geq 1$.

Thus, $\gcd(x^q-x, f)$ is the product of all the distinct linear factors of $f$. And so, if $f$ has no linear factors, then $\gcd(x^q-x , f )$ is the product of all the distinct quadratic irreducible factors of $f$. And so on.

I was wondering that there is a way to generalize that result to a polynomial in 2 or more variables.

Thank you

• I seriously doubt it. Things get incredibly more complicated going from one dimension to two dimensions. – Gregory Grant Feb 2 '16 at 22:06
• Indeed, I fund some research in the subject, but they generated very dense and not intuitive algorithms for test irreducibility in multivariables polynomials. I understand it is no an easy problem, but my ask is there is not a simpler criterion to check for irreducibility of multivariate polynomials? – Luis GC Feb 17 '16 at 20:59