Methods to see if a polynomial is irreducible Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now. First is Eisenstein criterion. Another is that if a polynomial is irreducible mod p then it is irreducible. Are there any others?

Here's an elementary trick that I occasionally find useful: Let $y=x+c$ for some fixed integer $c$, and write $f(x)=g(y)$. Then $f$ is irreducible if and only if $g$ is irreducible. You may be able to able to reduce $g$ modulo a prime and/or apply Eisenstein to show that $g$ is irreducible.
