  The polynomial $x^n+1$ over finite fields has been of interest due to its
applications in the study of negacyclic codes over finite fields. In this
paper, a rigorous treatment of the factorization of $x^n+1$ over finite fields
is given as well as its applications. Explicit and recursive methods for
factorizing $x^n+1$ over finite fields are provided together with the
enumeration formula. As applications, some families of negacyclic codes are
revisited with more clear and simpler forms.
