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?

**Answer**

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 [1] on this topic.

[1] http://www.math.sc.edu/~filaseta/gradcourses/Math788F/latexbook/

[2] http://www.math.sc.edu/~filaseta/gradcourses/Math788F/NewtonPolygonsTalk.pdf

[3] Newton Polygon Applet

http://www.math.sc.edu/~filaseta/newton/newton.html

[4] Abhyankar, Shreeram S.

Historical ramblings in algebraic geometry and related algebra.

Amer. Math. Monthly 83 (1976), no. 6, 409-448.

http://links.jstor.org/sici?sici=0002-9890(197606/07)83:6%3C409:HRIAG…

**Attribution***Source : Link , Question Author : Community , Answer Author : Bill Dubuque*