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Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In thisarticle, 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 explicitfactorization 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 isgiven.

Explicit factorization of $x^n-1\in \mathbb F_q[x]$