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?

Solution 1:

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...

Solution 2:

One method for polynomials over $\mathbb{Z}$ is to use complex analysis to say something about the location of the roots. Often Rouche's theorem is useful; this is how Perron's criterion is proven, which says that a monic polynomial $x^n + a_{n-1} x^{n-1} + ... + a_0$ with integer coefficients is irreducible if $|a_{n-1}| > 1 + |a_{n-2}| + ... + |a_0|$ and $a_0 \neq 0$. A basic observation is that knowing a polynomial is reducible places constraints on where its roots can be; for example, if a monic polynomial with prime constant coefficient $p$ is reducible, one of its irreducible factors has constant term $\pm p$ and the rest have constant term $\pm 1$. It follows that the polynomial has at least one root inside the unit circle and at least one root outside.

An important thing to keep in mind here is that there exist irreducible polynomials over $\mathbb{Z}$ which are reducible modulo every prime. For example, $x^4 + 16$ is such a polynomial. So the modular technique is not enough in general.

Solution 3:

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.