Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR（pe,d） is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f（x） the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois...

Full description

Saved in:
Bibliographic Details
Published in:Science China. Mathematics 2013-09, Vol.56 (9), p.1761-1772
Main Authors: Zhang, XiaoLei, Hu, Lei
Format: Article
Language:English
Subjects:
Applications of Mathematics
Astronomy
China
Factorization
Galois扩张
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Residues
Rings (mathematics)
Series (mathematics)
Upper bounds
X-1
上限值
不可约多项式
伽罗瓦
概率
聚乙烯
表达式
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The period of a monic polynomial over an arbitrary Galois ring GR（pe,d） is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f（x） the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f（x） is characterized by the periods of the irreducible polynomial and an associated polynomial of the form （x-1）m＋pg（x）. Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR（pe,d） is equal to pe-1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.
ISSN:1674-7283
1006-9283
1869-1862
DOI:10.1007/s11425-013-4592-2