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