Factorization of composed polynomials and applications

2019/01/09 by Fabio Enrique Brochero Martínez, Martínez, F. E. Brochero, Lucas Reis +3 · 2 citations
Computer Science · #12E20 (primary) and 11T30 (secondary) #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1901.02951

openalex publication_date 2019/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF_q be the finite field with q elements, where q is a prime power and n be a positive integer. In this paper, we explore the factorization of f(xⁿ) over \mathbbF_q, where f(x) is an irreducible polynomial over \mathbb F_q. Our main results provide generalizations of recent works on the factorization of binomials xⁿ-1. As an application, we provide an explicit formula for the number of irreducible factors of f(xⁿ) under some generic conditions on f and n.

Cited by

• The 3-sparsity of Xⁿ-1 over finite fields, II
• The 3-sparsity of Xⁿ-1 over finite fields

Related

• Explicit factorization of xⁿ-1∈ \mathbb F_q[x]
• Further factorization of xⁿ-1 over a finite field
• Further factorization of xⁿ-1 over finite fields (II)
• Factorization of a class of composed polynomials
• Factoring polynomials of the form f(xⁿ)∈ \mathbbF_q[x]