A fatoração de xⁿ-1 em binômios e trinômios irredutíveis sobre os corpos com 2 e 4 elementos

Abstract

Let q be a prime power and Fq the finite field with q elements. The factorization of polynomials over finite fields plays a fundamental role in several areas, such as error-correcting coding theory and cryptography. A particularly important polynomial is xⁿ-1, since, for example, each of its irreducible factors corresponds to a cyclic code. Under certain conditions, the polynomial xⁿ-1 factors exclusively into irreducible binomials and trinomials over Fq. In this case, we say that the pair (n, q) is 3-sparse. The aim of this work is to give a complete classification of the pairs (n, 2) and (n, 4) that are 3-sparse.