On Primitive Optimal Normal Elements of Finite Fields

DOI：10.3770/j.issn:1000-341X.2010.05.014

 作者 单位 廖群英 四川师范大学数学与软件科学学院, 四川 成都 610066

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.