# $(X^2+1)(X^4+X^2+X+1)$ factorizes as product of degree $1$ polynomials over $81$ elements field

I've been solving problems from my Galois Theory course, and I want to check if my approach to this one is correct. It says:

Check if the polynomial $$(X^2+1)(X^4+X^2+X+1)$$ factorizes as product of degree $$1$$ polynomials in a field with $$81$$ elements.

What I've done is notice that, if the field has exactly $$81=3^4$$ elements, then it's equal to $$\mathbb Z_3(\alpha)$$, where $$\alpha$$ is a root of the polynomial $$X^{3^4}-X$$. Given that every finite extension of finite fields is normal, then $$\mathbb Z_3(\alpha)$$ is also normal, so the question is equivalent to prove if $$X^2+1$$ and $$X^4+X^2+X+1$$ divide $$X^{3^4}-X$$ or not.

Considering now that if $$d\mid n$$, then every irreducible polynomial of degree $$d$$ in $$\mathbb Z_p[X]$$ divides the polynomial $$X^{p^n}-X$$ in $$\mathbb{Z}_p[X]$$ (it's stated inside my course notes), in my particular case $$n=4$$, so from the fact that $$2\mid 4$$ and $$4\mid 4$$ we conclude $$X^2+1\mid X^{3^4}-X$$. Also, $$X^4+X^2+X+1$$ can either be irreducible or decompose as product of degree 2 irreducible polynomials, so in any case also $$X^4+X^2+X+1\mid (X^{3^4}-X)$$.

Finally, since our 81 elements field is the splitting field of $$X^{3^4}-X$$ over $$\mathbb{Z}_3$$, it follows that $$(X^2+1)(X^4+X^2+X+1)$$ factorizes as degree 1 polynomials over our field.

Is my work correct? If not, where did I go wrong? Any help will be appreciated, thanks in advance.

• @ancientmathematician: The other factor is not $(X^5-1)/(X-1)=X^4+X^3+X^2+X+1$. Jun 8, 2021 at 13:40
• @MarcvanLeeuwen You are so right. I am embarassed. But I bet it was meant to be. ;-) Jun 8, 2021 at 13:44
• @ancientmathematician it was not meant to be, it was just $X^4+X^2+X+1$. No worries! thanks for your help. Jun 8, 2021 at 14:06

Your argument is almost complete. You know that all irreducible polynomials (over $$\Bbb Z/3\Bbb Z$$) of degree $$1$$, $$2$$ or $$4$$ split into linear factors in a field with $$81$$ elements. So your only worry is that the polynomial $$X^4+X^2+X+1$$ might have an irreducible factor of degree$$~3$$, which would then fail to factor. The remaining factor after dividing by that factor of degree$$~3$$ would be of degree$$~1$$ and give a root of $$X^4+X^2+X+1$$ in $$\Bbb Z/3\Bbb Z$$. But none of the values $$0,1,2$$ are roots, so this does not happen. (With a bit more effort you can see that the polynomial $$X^4+X^2+X+1$$ over $$\Bbb Z/3\Bbb Z$$ is in fact irreducible).