| 董可静,曹喜望.有限域上幂剩余正规元的存在性[J].数学年刊A辑,2011,32(3):365~374 |
| 有限域上幂剩余正规元的存在性 |
| On the Existence of Power Residual Normal Elements Over Finite Fields |
| |
| DOI： |
| 中文关键词: 有限域 幂剩余 正规元 指数和 |
| 英文关键词:Finite field Power residues Normal elements Exponential sums |
| 基金项目:国家自然科学基金(No.10971250)资助的项目 |
| Author Name | Affiliation | E-mail | | DONG Kejing | Department of Mathematics,Nanjing University of Aeronautics and Astronautics,Nanjing 211100,China. | dongkejing.s@126.com | | CAO Xiwang | Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China
Department of Mathematics, LMIB of Ministry of Education,
Beijing University of Aeronautics and Astronautics, Beijing 100191, China
State Key Laboratory of Information Security, Graduate University of Chinese Academy of Sciences,
Beijing 100049, China. | xwcao@nuaa.edu.cn |
|
| Hits: 1196 |
| Download times: 722 |
| 中文摘要: |
| ${\rm GF}(q)$是$q$个元的有限域, $q$是素数的方幂, $n$是正整数, ${\rm GF}(q^n)$为${\rm GF}(q)$的$n$次扩张. 用指数和估计的方法给出了3种情形下幂剩余正规元存在的充分条件,
即 (1) ${\rm GF}(q^n)$中存在元$\xi$为${\rm GF}(q)$上的幂剩余正规元;
(2) ${\rm GF}(q^n)$中存在元$\xi$与$\xi^{-1}$同时为${\rm GF}(q)$上幂剩余正规元;
(3) 对${{\rm GF}(q^n)}^*$中任意
给定的非零元$a$和$b$, ${\rm GF}(q^n)$中存在元$\xi$与$\xi^{-1}$同时为${\rm GF}(q)$上$d$次幂剩余正规元, 且满足${\rm Tr}(\xi)=a$, ${\rm Tr}(\xi^{-1})=b$. |
| 英文摘要: |
| Let ${\rm GF}(q)$ denote a finite field with $q$ elements,
$q$ be a prime power, $n$ a positive integer, and ${\rm GF}(q^n)$ the $n$-th Galois
extension of ${\rm GF}(q)$.
By using exponential sums, some sufficient conditions are given
for the existence of certain power residual normal elements in the
following three cases: (1) $\xi$ ${\in \rm GF}(q^n)$ is a power residual normal element in ${\rm GF}(q)$;
(2) Both $\xi$ and $\xi^{-1}$ ${\in \rm GF}(q^n)$ are power residual
normal elements in ${ \rm GF}(q)$; (3) For arbitarily given two elements $a,b$ in ${{\rm GF}(q^n)}^*$, the existence of the element $\xi$ such that both
$\xi$ and $\xi^{-1}$ are $d$-th power residual normal elements in ${ \rm GF}(q)$ satisfying ${\rm Tr}(\xi)=a$, ${\rm Tr}(\xi^{-1})=b$. |
| View Full Text View/Add Comment Download reader |
| Close |
|
|
|
|
|
