Checking irreducibility

Dr. Kim

Would you mind telling me the different methods i can use to prove polynomial irreducibility over Q. I know Eisenstein’s Criterion and the brute force approach of finding no zeros that divide the constant (gauss?).

Also i was confused about this. Homework Sheet 2, question 2:

Show f(X)=X^3 - 2 is irreducible over Q

In the solutions it says irreducible by eisenstein p=2 but i thought p^2=4 and (-2)+4=2=0 \mod 2 ? Am i correct or have i misunderstood the criterion. Instead is it ok to show +-1,+-2 are not zeros of the polynomial?

Thanks for your help, i thought i would ask these basic questions now as not to waste anyone’s time at our revision class.

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Reply:

I won’t explain in much detail and the general problem can be quite complex, although there are various algorithms for dealing with it.

The facts you refer to are:

1. f(x) has a linear factor if and only if it has a rational root. Look at the online notes lecture 1 for the proof and for a review of how to check for rational roots. A corollary of this is that a polynomial f(x) of degree two or three is irreducible if and only if it has no roots. (This statement, by the way, is true over any field.)

2. Suppose f(x) has degree 4. Then the possibilities for reducibility are:

(a) f(x)=g(x)h(x)

where g has degree 1 and h has dergee 3;

(b) g, h both have degree two. To consider this possibility is slightly more tedious, but still manageable by hand.

To consider the general possibilities in this manner for polynomials of degree five or more becomes quite unwieldy.

But in special situations, one can use

3. Eisenstein’s criterion. Perhaps you have misunderstood the statement of the criterion. In the example you mention, since the prime 2 divides all non-leading coefficients and 2^2 does not divide the constant term, the criterion says x^3-2 is irreducible. Since this polynomial has degree 3, you are right that one could simply check that it has no rational roots to conclude irreducibility. But Eisenstein’s criterion is already useful for x^4-2 since we don’t have to go through the tedious check to rule out degree two factors.

4. It is often convenient to use a change of variable:

f(x) is irreducible if and only if f(ax+b) is irreducible for some constants a,b. Try writing out for yourself the elemenary proof of this. As an application, we checked, for example, the irreducibility of

f(x)=x^{p-1}+x^{p-2}+\cdots+x+1

for any prime p by checking the irreducibility of f(x+1) using Eisenstein’s criterion.

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