有限域的原根在剩余理論中的應(yīng)用

孫翠芳,程智

(安徽師范大學(xué)數(shù)學(xué)系,安徽蕪湖 241000)

有限域的原根在剩余理論中的應(yīng)用

孫翠芳,程智

(安徽師范大學(xué)數(shù)學(xué)系,安徽蕪湖 241000)

研究了p是正奇素?cái)?shù)的情況下,有限域的原根在剩余理論中的應(yīng)用.利用有限域Fp上原根的性質(zhì),給出了一類集合與平方剩余之間的關(guān)系,獲得了這類集合所包含元素個(gè)數(shù)之間的關(guān)系,并且這個(gè)結(jié)論把關(guān)于這方面的結(jié)果從r=0推廣到了r=0和2.

有限域;原根;平方剩余;四次剩余特征

1 引言

其中?x?表示不超過(guò)x的最大整數(shù).

利用原根和子群生成元的性質(zhì),在文[3-4]中,孫智宏證明了:

定理1.1設(shè)g是Fp的原根,則

2 幾個(gè)引理

3 本文主要結(jié)果

[1]楊明順.關(guān)于偽Smarabdache函數(shù)的一個(gè)問(wèn)題[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2008,24(3):449-451.

[2]Ireland K,Rosen M.A Classical Introduction to Modern Number Theory[M].2nd ed.New York:Springer-Verlag,1990.

[3]SUN Zhihong.Consecutive numbers with the same Legendre symbol[J].Proceedings of Amer.Math.Soc., 2002,130(9):2503-2507.

[4]SUN Zhihong.Supplements to the theory of quartic residues[J].Acta Arith.,2001,97:361–377.

[5]Kenneth Rosen H.Elementary Number Theory and Its Applications[M].5th ed.Reading Massachusetts: Addison Wesley,2000.

[6]Berndt B C,Evans R J,Williams K S.Gauss and Jacobi Sums[M].New York:Wiley,1998.

The applications of primitive roots of finite field to the theory of residues

SUN Cui-fang,CHENG Zhi
(Department of Mathematics,Anhui Normal University,Wuhu241000,China)

The paper discusses the applications of primitive roots of finite fields to the theory of residues with a positive odd prime p.The relations between some sets and quadratic residue are given by using the properties of primitive roots of the finite field Fp,also we give relations of the number of elements in the sets,the conclusions generalize the results from the case of r=0 to r=0,2.

finite fields,primitive roots,quadratic residue,biquadratic residue characters

O156.2

A

1008-5513(2009)03-0486-05

2008-10-18.

國(guó)家自然科學(xué)基金(10071001),安徽省自然科學(xué)基金(01046103),安徽省教育廳自然科學(xué)基金(2002KJ131),校青年基金(2005xqn04,2007xqn52).

孫翠芳(1978-),碩士,講師,研究方向：計(jì)算數(shù)論及其應(yīng)用.

2000MSC:11A07,11A15