Determining Irreducibility of Polynomials

[pivoxa15]

Homework Statement

What are the ways you can determine whether a polynomial is irreducible?

The Attempt at a Solution

Eisenstein's criterion is one but it can't be applied all the time. i.e. x^4-2x^2+9

 

[Hurkyl]

Off the top of my head, I know two things you can do:

(1) See if it's irreducible over several small finite fields. For your polynomial, that just means checking that GCD(x^4 - 2x^2 + 9, x^(p^k) - x) = 1, for k = 1, 2, and 3.

(2) Compute a root of the polynomial and determine its minimal polynomial. (maybe by computing it numerically over the complexes and using lattice basis reduction to find the minimal polynomial exactly, or maybe by solving it over the 2-adics, or maybe by solving it modulo several small primes and using the CRT)