I'm asking this question with regards to when a polynomial $m(t)$ can be the minimal polynomial of an algebraic element $\alpha$ of a field extension $F(\alpha):F$, when $F$ is a finite field.
For example, if we have a polynomial like $t^{37}+1$, we know the polynomial has an obvious root in $\mathbb{F}_p$, namely we can use $(p-1)^{37}+1=0$. That means we can factorise the polynomial into $t^{37}+1=(t-(p-1))\cdot(t^{36}+\ldots+(p-1))$. What about general monic polynomials over finite fields? Is there something analogous to Eisenstein's criterion that we can use to check for irreducibility?
If the polynomial is reducible in $\mathbb{Z}[x]$, then it is reducible over $\mathbb{F}_p[x]$ for every prime. The converse unfortunately is not true, e.g., $x^4+1$ is irreducible in $\mathbb{Z}[x]$, but reducible over any finite field. On the other hand, if we have a given polynomial in $\mathbb{F}_q[x]$, we can use several algorithms, e.g., the Berlekamp algorithm, to obtain a factorisation into powers of irreducible polynomials. For example, $$ x^{37} + 37x + 37 $$ is irreducible over a finite field with $103$ elements, but reducible over a field with $101$ elements, namely $$ x^{37} + 37x + 37=(x+17)(x^2+9x+99)(x^{34}+ \cdots +91). $$