A Note on Factorization and the Number of Irreducible Factors of x n − λ over Finite Fields

Let F q be a finite field, and let n be a positive integer such that gcd ( q , n ) = 1 . The irreducible factors of x n − 1 and x n − λ are fundamental concepts with wide applications in cyclic codes and constacyclic codes. Furthermore, the number of irreducible factors of x n − 1 and x n − λ is useful in many computational problems involving cyclic codes and constacyclic codes. In this paper, we give a more concrete irreducible factorization of x n − 1 and x n − λ . Based on this, the number of irreducible factors of x n − 1 and x n − λ over F q , for any λ ∈ F q ∗ , is determined through research on the representatives and the sizes of the q -cyclotomic cosets. As applications, we present the necessary and sufficient conditions for ∣ F ( x n − 1 ) ∣ = 6 and a more concrete factorization of x n − 1 in these cases.