Further factorization of xⁿ-1 over a finite field

2017/10/22 by Yansheng Wu, Qin Yue, Wu, Yansheng +3
Computer Science · Social Sciences · #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Islamic Finance and Communication #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1710.07943

openalex publication_date 2017/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \Bbb F_q be a finite field with q elements and n a positive integer. Martínez, Vergara and Oliveira \citeMVO explicitly factorized xⁿ - 1 over \Bbb F_q under the condition of rad(n)|(q-1). In this paper, suppose that rad(n)\nmid (q-1) and rad(n)|(q^w-1), where w is a prime, we explicitly factorize xⁿ-1 into irreducible factors in \Bbb F_q[x] and count the number of its irreducible factors.

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