Given a finite field $\mathbb{F}_p$ and some polynomial $f(x)\in\mathbb{F}_p [x]$. What are some of the methods of determining the irreducibility of $f(x)$? I feel like there are many theorems that we can use if we are trying to determine irreducibility over $\mathbb{Q}[x]$.
When $p$ and $\deg(f)$ are small, we can compute directly. For example, if we have something like $f(x) = x^4 + x^3 + 1$ in $\mathbb{F}_2 [x]$ or $\mathbb{F}_4[x]$, by plugging in we know that $f$ has not root in both of them. But what if I have something like $\mathbb{F}_{64}$ or $\deg(f)$ being very big?