Factorize certain polynomials as product of irreducibles

Question

I'm trying to solve this question from my abstract algebra's course. I know it's a very trivial question that can be solved using Ruffini like we are taught before college, but my question is how can I solve it using "ring of polynomial" theory. The question goes like this:

Factorize the following polynomials as product of irreducibles:

1. $X^5 − 2X^4 − 12X + 24$ in $\mathbb{Z}[X]$
2. $X^4+X^2+1$ in $(\mathbb{Z}/2\mathbb{Z})[X]$

I know about ring theory and some ring of polynomials theory, but I don't see what can I do here to approach this kind of problem using abstract algebra methods (instead of Ruffini, because I guess I'm not supposed to solve it that way after all the theory I've studied, where this method is never introduced). How can I solve this? Any help will be appreciated, thanks in advance.

Answer 1

1. $2$ is an obvious root of $P(X)=X^5 − 2X^4 − 12X + 24$, hence $P(X)=(X-2)Q(X)$. Try to apply Eisenstein Criterion to $Q(X)$.
2. $\mathbf Z/2\mathbf Z$ has characteristic $2$, hence $\mathbf Z[X]$ too, so that $x\mapsto x^2$ is a ring homomorphism (the Frobenius homomorphism).

Answer 2

In view of $GF(2)[x]$ with $GF(2) = \Bbb Z/2\Bbb Z$, the irreducible polynomials of degree at most 4 are

$x,x+1,x^2+x+1,x^3+x+1,x^3+x^2+1, x^4+x+1, x^4+x^3+1, x^4+x^3+x^2+x+1$.

This should suffice to factorize $x^4+x^2+1$ which is reducible.