论文标题
$ k $ - 正常元素在有限领域的存在和心脏
Existence and Cardinality of $k$-Normal Elements in Finite Fields
论文作者
论文摘要
有限领域的正常基础构成了一个庞大的理论和实际兴趣的广泛主题。最近，引入了$ k $正常的元素，作为普通元素的自然扩展。在有限领域的固定扩展中的存在和$ k $正常的元素的数量既是一般性的开放问题，又包括一个有希望的研究途径。在本文中，我们首先为$ k $ - 正常元素的数量制定了一般的下限，假设它们存在。我们进一步通过多项式$ x^m-1 $的一般分解为$ k $ normal元素的新存在条件，用于环形多项式。最后，我们在$ \ fqm $中为普通元素提供了一个存在条件，并在有限字段的单元组中具有非最大值但高的乘法顺序。
Normal bases in finite fields constitute a vast topic of large theoretical and practical interest. Recently, $k$-normal elements were introduced as a natural extension of normal elements. The existence and the number of $k$-normal elements in a fixed extension of a finite field are both open problems in full generality, and comprise a promising research avenue. In this paper, we first formulate a general lower bound for the number of $k$-normal elements, assuming that they exist. We further derive a new existence condition for $k$-normal elements using the general factorization of the polynomial $x^m-1$ into cyclotomic polynomials. Finally, we provide an existence condition for normal elements in $\fqm$ with a non-maximal but high multiplicative order in the group of units of the finite field.
