有限域上的本原最优正规元
On Primitive Optimal Normal Elements of Finite Fields
设q为素数的幂,F_{q^{n}}是q元有限域F_{q}的n(n>1)次扩张,Davenport,Lenstra以及Schoof等人曾证明了:存在F_{q^{n}}中的本原元素alpha使得alpha生成F_{q^{n}}在F_{q}上的一组正规基。随后,Mullin, Gao以及Lenstra等人,提出了最优正规基的概念并给出了这种正规基的构造,本文给出了全部的本原I型最优正规基,以及所有这样的有限扩域F_{q^{n}}/F_{q}: F_{q^{n}}中存在一对互逆的元素alpha,alpha^{-1}使得alpha和alpha^{-1}均生成F_{q^{n}}在F_{q}上的最优正规基。最后,我们给出了本原II型最优正规基存在的一个充分条件,并且证明了所有的本原最优正规元是彼此共轭的。
Let q be a prime or prime power and F_{q^{n}} the extension of q elements finite field F_{q} with degree n(n>1). Davenport, Lenstra and Schoof proved that there exists a primitive element \\alpha\\in F_{q^{n}}such that \\alpha generates a normal basis of F_{q^{n}} over F_{q}. Later, Mullin, Gao and Lenstra, etc., raised the definition of optimal normal bases and constructed such bases. In this paper, we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements \\alpha and \\alpha^{-1} such that both of them generate optimal normal bases of F_{q^{n}} over F_{q}. Furthermore, we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.
廖群英
数学
有限域正规基本原元最优正规基
Finite fieldsNormal basesPrimitive elementsOptimal normal bases
廖群英.有限域上的本原最优正规元[EB/OL].(2010-01-19)[2025-08-02].http://www.paper.edu.cn/releasepaper/content/201001-806.点此复制
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