TITLE: Is the polynomial $x^{105} - 9$ reducible over $\mathbb{Z}$?
QUESTION [9 upvotes]: Is the polynomial $x^{105} - 9$ reducible over $\mathbb{Z}$?

This exercise I received on a test, and I didn't resolve it. I would be curious in any demonstration with explanations. Thanks!

REPLY [2 votes]: For binomials there is a classical irreducibility test (below). It implies that $\,x^{105}-c\,$ is irreducible over a field $\,F\,$ if $\,c\,$ is not a third, fifth, or seventh power in $\,F,$ since $\,105 = 3\cdot 5\cdot 7.$
Theorem $\ $ Suppose $\:c\in F\:$ a field,  and $\:0 < n\in\mathbb Z.$
$\quad x^n - c\ $ is irreducible over $\:F \iff c \not\in F^p\:$ for all primes $\,p\mid n\:$ and $\ c\not\in -4F^4$ when $\: 4\mid n$.
Proofs can be found in many Field Theory textbooks, e.g. see Lang's Algebra, or see Karpilovsky, Topics in Field Theory, Theorem 8.1.6.