  Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In this
article, we prove that, under some conditions on $q$ and $n$, the polynomial
$x^n-1$ can be split into irreducible binomials $x^t-a$ and an explicit
factorization into irreducible factors is given.
  Finally, weakening one of our hypothesis, we also obtain factors of the form
$x^{2t}-ax^t+b$ and explicit splitting of $x^n-1$ into irreducible factors is
given.
