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数学进展 - 2018, Vol. 47(2): 161-174
研究论文
$x^{N} \pm a$在有限域上的完全分解
Explicit Factorization of \mbox{\boldmath $x^{N} \pm a$} over a Finite Field

王玉琨*,曹喜望
WANG Yukun, CAO Xiwang

南京航空航天大学理学院, 南京, 江苏, 211106
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211106, P. R. China

出版日期: 2018-05-16
2018, Vol. 47(2): 161-174
DOI: 10.11845/sxjz.2016094b


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摘要 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$N=2^{m}p^{n}$时$x^{N} \pm a\in F_q[x]$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$p\neq 2$. 结果表明当$a$取作$F_q$中元素$\beta$的某些特殊方幂时, $x^{N} \pm a$在$F_q$上不可约因式都是二项式或三项式.
关键词 完全分解二项式三项式有限域    
Abstract:Let $F_q$ denote a finite field of {\rm ord}er $q$, 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 $x^{N}\pm a$ over $F_q$ is given in this paper, where $N=2^{m}p^{n}$, $m$, $n$ are positive integers, $a \in F_q$, $p$ is odd prime divisor of $q-1$. The results show that when $a$ is certain power of an element $\beta \in F_q$, the irreducible factor of $x^{N}\pm a$ in $F_q$ is either a binomial or a trinomial.
Key wordsexplicit factorization    binomial    trinomial    finite field
PACS:  O157.4  
基金资助:南京航空航天大学研究生创新基地(实验室)开放基金(No. kfjj20160802)和中央高校基本科研业务费专项资金资助.
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