Find all irreducible factors of a polynomial over $\mathbb{Z}$

Question

I am trying to solve this question for my abstract algebra class.

Let $f = X^6 - 2X^5 + 3X^4 - 2X^3 + 3X^2 - 2X + 2 \in \mathbb{Z}[X]$.

Either prove that $f$ is irreducible over $\mathbb{Z}$ or find all irreducible factors of $f$.

I know the answer is $(X^2 - 2X + 2)(X^2 - X + 1)(X^2 + X + 1)$.

However, I have no idea how we came up with this factorization. I have read this post, but I don't understand:

1. Why we assume the factors are monic polynomials?
2. Can we apply this to all kinds of polynomials, over any field?

I would also appreciate any different point of view, or any different solution.

Answer 1

By the Rational Root Theorem there is no linear factor. For testing quadratic factors we expand $(x^4+ax^3+bx^2+cx+d)(x^2+rx+s)$ and compare the coefficients with $f$. This gives easy equations in $a,b,c,d,r,s\in \Bbb Z$. Note that we obtain for example $ds=2$, which implies up to signs either $d=1$ and $s=2$ or vice versa. This is one reason that we can solve these equations easily. Finally, we repeat this for the quartic polynomial. So we obtain the complete factorization.