有限域上某些特殊元素的存在性问题  

Existenceof Some Specific Elements in FiniteFields

作　　者：张杭隆 曹喜望[1,2] Hang Long ZHANG;Xi Wang CAO(Department of Mathematics,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,P.R.China;Key Laboratory of Mathematical Modelling and High Performance Computing of AirVehicles(NUAA),MIIT,Nanjing 211106,P.R.China)

机构地区：[1]南京航空航天大学数学系,南京211106 [2]飞行器数学建模与高性能计算工业和信息化部重点实验室(南京航空航天大学),南京211106

出　　处：《数学学报（中文版）》2022年第6期1023-1032,共10页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11771007,12171241,61572027)。

摘　　要：设q=p^(k)且F_(qn)是q元有限域F_(q)的n次扩张,其中p为奇素数且n,k都是正整数.本文主要结论是:若n|(q-1),k≥11且n≥14或n1|(q-1),k≥10且n≥8,则F_(qn)中存在本原元α,使得α+α^(-1)为正规元且1+α^(2)为平方元,同时存在正规元β,使得β+β^(-1)为本原元且1+β^(2)为平方元.Let q=p^(k) and Fqn be the extension field of F_(q) of degree n,where p is an odd prime and n,k are positive integers.The main contribution of this paper is as follows:If nl(q-1),k≥1l,n≥14 or n t(q-1),k≥10,n≥8,then there exists a primitive elementαin Fqn such thatα+α^(-1) is a normal element,and 1+α^(2) is a square element,and there exists a normal elementβ,such thatβ+β^(-1) is a primitive element,and 1+β^(2) is a square element.

关 键 词：有限域 本原元 正规元 平方元 

分 类 号：O153.4[理学—数学]