Further factorization of xn − 1 over a finite field

Abstract

Let Fq be a finite field with q elements and n a positive integer. Martínez, Vergara and Oliveira (Des. Codes Cryptogr. 77 (1): 277–286, 2015) explicitly factorized xn−1 over Fq under the condition of rad(n)|(q−1). In this paper, we factorize xn−1 into irreducible factors in Fq[x] and count the number of its irreducible factors when rad(n)∤(q−1), rad(n)|(qw−1), and w is a prime.