Factorization m(x) over Q

Question

I need to factorize $m(x) = x^q-1$ (q is odd prime number) over $\mathbb{Q}$. I have filed $K = \mathbb{Q}$ and polynomial $m \in K[x]$. Could someone help me, how to do it?

Answer 1

$\dfrac{(x+1)^q-1}{x}$ satisfies Eisenstein's criterion: All coefficients (except the leading term) are divisible by $q$ and the constant term is not divisible by $q^2$ (because it is equal to $q$). Therefore, it is irreducible.

Therefore $\dfrac{x^q-1}{x-1}$ is irreducible too.