Factoring quadratic polynomials over finite fields can easily be done by determining if the discriminant is a quadratic residue modulo characteristic in question, and if so, apply the quadratic formula to obtain a solution in that finite field. Otherwise, the quadratic remains irreducible over that field. This characterizes behaviors of quadratic polynomials factoring over finite fields.
What about cubic polynomials, particularly those whose Galois group is isomorphic to the symmetric group $S_3$?
For example, I want to determine the factoring behavior of $f(x)=x^3-x-1$ over $\mathbb{F}_p$. I know that $f$ is the equation for the Hilbert Class Field of $\mathbb{Q}(\sqrt{-23})$.
Hence, I would assume that for all primes which can be represented as $p = 23x^2+y^2$ for integers $x,y$, then $f$ will decompose into linear factors over $\mathbb{F}_p$. Hence, the splitting field of $f$ over $\mathbb{F}_p$ is itself.
In the case that $-23$ is a quadratic non-residue $\pmod p$, then $f$ decomposes into one linear factor and one quadratic factor over $\mathbb{F}_p$. Hence, $\mathbb{F}_{p^2}$ is the splitting field of $f$ over $\mathbb{F}_p$.
Lastly, if $-23$ is a quadratic residue $\pmod p$, but $p\neq 23x^2+y^2$ for integers $x, y$, then $f$ should be irreducible over $\mathbb{F}_p$, so the splitting field becomes $\mathbb{F}_{p^3}$.
I've made some observations here, but how would I go about proving them? Any help is appreciated! Thanks!