Further factorization of x^n-1 over finite fields (II)

12/15/2020

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by   Yansheng Wu, et al.

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