Decompose in irreducible factors over $\mathbb{Z}_3[X]$

Question

I've been solving some problems from my Galois Theory course, but I can't solve this one and don't find help in my course notes. It says:

Decompose in product of irreducible factors over $\mathbb{Z}_3[X]$ the following polynomials

1. $X^{26}-1$
2. $X^7+X^6+X^5+X^4+X^3+X^2+X+1$
3. $X^8-X^7+X^5-X^4+X^3-X+1$

The work I've done so far:

1. For this first one, it was clear $1$ was a root, so one irreducible factor is $(X-1)=(X+2)$ and using cyclotomic equation I can write it as $$X^{26}-1=(X-1)(X^{25}+X^{24}+X^{23}+\cdots+X+1).$$ My problem now is that I don't know how to find the irreducible factors of the right term (I guess it's not irreducible, maybe I'm wrong, in that case how do I prove its irreducibility?).
2. For this one, I first found that $2$ is root. I did it by using the Frobenius automorphism that goes from $x$ to $x^3$, doing the following: $$X^3X^3X+X^3X^3+X^3X^2+X^3X+X^3+X^2+X+1=$$ $$=XXX+ XX + XX^2+XX+X+X^2+X+1=$$ $$X+X^2+X+X^2+X+X^2+X+1= 3X^2+3X+X+1=X+1$$ so $2$ is root and one irreducible factor is $(X-2)=(X+1)$. Then I used Ruffini to divide the original polynomial and got that it's equal to $$(X-2)(X^6+X^4+X^2+1),$$ and I don't know how to continue from this point.
3. For this last one, I tried something similar to what I did in point 2, but got no results (Frobenius showed the polynomial is always 1 after evaluating in one element of $\mathbb{Z}_3$), so for this one I don't know where to start.

Is the work I've already done correct? If not, where did I go wrong? How can I finish this problem? Any help will be appreciated, thanks in advance.

Answer 1

(1,2): Note they are basically asking for $X^{3^n-1}-1$, and what do you know about $\mathbb{F}_{3^n}^\times$?

(3) Note you could use characteristic 3 to jump immediately to $(X^4+X^3+X^2+X+1)^2$ (since its coefficients are 1,2,3,4,5,4,3,2,1 over $\mathbb{Z}$), or arrive at it by computing $\gcd(f,f')$