x^N ±a在有限域上的完全分解

Explicit Factorization of x^N ±a over a Finite Field

设F_q为一个阶为q的有限域,其中q为奇素数的幂.本文主要利用多项式分解相关理论得到几类多项式的完全分解,给出了当N=2~mp^n时x^N±a∈F_q[x]在F_q上的完全分解,其中m,n均为正整数,p为q-1的素因子,且p≠2.结果表明当a取作F_q中元素β的某些特殊方幂时,x^N±a在F_q上不可约因式都是二项式或三项式.

Let Fq denote a finite field of order q, where q is the power of odd prime. In this paper, we use some results on the factorizations of polynomial to get the explicit factorization of a class of polynomials. The explicit factorization of X^N±α over Fq is given in this paper, where N = 2m^p^n, m, n are positive integers, X^N±α, p is odd prime divisor of q - 1. The results show that when a is certain power of an element β ∈ Fq, the irreducible factor of X^N±α in Fq is either a binonmial or a trinomial.