多项式x^n-1 在有限域上的分解

Factorization of polynomial x^n-1 over finite fields

摘要:

多项式 x^n-1 在有限域 F_q 上的分解不仅在理论上有重要意义, 在保密通信、纠错码等诸多方面也有深入应用. 本文在 ord_{rad(n)}q=2w (w 为奇素数)时, 得到了 x^n-1 的全部不可约因式, 这些结果部分完善和推广了近期相关方面的研究成果.

Abstract:

Factorization of the polynomial x^n-1 over finite fields is not only important theoretically, but also has a lots of applications, especially in secure communication and error-correcting coding theory. In this paper, an explicit factorization of x^n-1 into irreducible factors over the field F_q is given when ord_{rad(n)}q=2w, where w is an odd prime number. These results improve and generalize some recent progresses.