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?
To better understand the Eisenstein and related irreducibility tests you should learn about Newton polygons. It’s the master theorem behind all these related results. A good place to start is Filaseta’s notes – see the links below. Note: you may need to write to Filaseta to get access to his interesting book  on this topic.
 Newton Polygon Applet
 Abhyankar, Shreeram S.
Historical ramblings in algebraic geometry and related algebra.
Amer. Math. Monthly 83 (1976), no. 6, 409-448.