$x^n - 1$ factors in finite field

Question

I am having some difficulty with some Galois problems and I was wondering if it is always true that

$x^n-1$ is a factor of $x^{nm}-1$ in a finite field,

and if not what conditions need to be imposed.

Thanks!

Answer 1

$y^m-1 = (y-1)(y^{m-1}+y^{m-2}+\cdots+y+1)$. Then take $y=x^n$.