The set of representatives and explicit factorization of xn − 1 over finite fields

Abstract

Let [Formula: see text] be a positive integer and let [Formula: see text] be a finite field with [Formula: see text] elements, where [Formula: see text] is a prime power and [Formula: see text]. In this paper, we give the explicit factorization of [Formula: see text] over [Formula: see text] and count the number of its irreducible factors for the following conditions: [Formula: see text] are odd and [Formula: see text]. First, we present a method to obtain the set of all representatives of [Formula: see text]-cyclotomic cosets modulo [Formula: see text], where [Formula: see text]. This set of representatives is then used to find the irreducible factors of [Formula: see text] and the cyclotomic polynomial [Formula: see text] over [Formula: see text]. The form of irreducible factors of [Formula: see text] is characterized such that the coefficients of these irreducible factors are followed by second-order linear recurring sequences.