Further factorization of xn − 1 over a finite field (II)

Abstract

Let [Formula: see text] be a finite field with [Formula: see text] elements. Let [Formula: see text] be a positive integer with radical [Formula: see text], namely, the product of distinct prime divisors of [Formula: see text]. If the order of [Formula: see text] modulo [Formula: see text] is either 1 or a prime, then the irreducible factorization and a counting formula of irreducible factors of [Formula: see text] over [Formula: see text] were obtained by Martínez, Vergara and Oliveira, Explicit factorization of [Formula: see text], Des. Codes Cryptogr. 77(1) (2015) 277–286 and Wu, Yue and Fan, Further factorization of [Formula: see text] over a finite field, Finite Fields Appl. 54 (2018) 197–215. In this paper, we explicitly factorize [Formula: see text] into irreducible factors in [Formula: see text] and calculate the number of the irreducible factors when the order of [Formula: see text] modulo [Formula: see text] is a product of two primes.