I want to study the factorization of a specific polynomial with coefficients over finite fields. Let $f\in \mathbb{Z}$ a polynomial, I define the polynomial class as follows: suppose $(\bar{f})\in \mathbb{Z}/p\mathbb{Z}$ such that its factorization is $\bar{f}(x)=\bar{f}_1(x)^{e_1}...\bar{f}_g(x)^{e_g}\in \mathbb{F}_p[x]$. Then I denote its class as $d_1^{e_1},...,d_g^{e_g}$, where $d_i$ is the degree of $\bar{f}_i(x)$. As an example, a polynomial of degree 2 may has class $(2)$, $(1^2)$ or $(1,1)$.
Is there any criteria to classify $3^\text{rd}$ degree polynomials related to Legendre Symbol and discriminant, or other methods?
I have noticed that exists a theorem, that maybe can help, but it isn't enough.
Theorem: Let $f(x)$ be a monic polynomial of degree $n$ with integral coefficients in a $p$-adic field $\mathbb{F}_p$. Assume that $\bar{f}(x)$ has no repeated roots. Let $r$ be the number of irreductible factors of $\bar{f}(x)$ over the residue class field. Then $r\equiv n\mod 2\;$ if and only if the discriminant is a square in $\mathbb{F}_p$.