  Let $\Bbb F_q$ be a finite field with $q$ elements and $n$ a positive
integer. Mart\'inez, Vergara and Oliveira \cite{MVO} explicitly factorized
$x^{n} - 1$ over $\Bbb F_q$ under the condition of $rad(n)|(q-1)$. In this
paper, suppose that $rad(n)\nmid (q-1)$ and $rad(n)|(q^w-1)$, where $w$ is a
prime, we explicitly factorize $x^{n}-1$ into irreducible factors in $\Bbb
F_q[x]$ and count the number of its irreducible factors.
