有限域中$k$ -正规元的存在性和基数
Existence and Cardinality of $k$-Normal Elements in Finite Fields
摘要：
有限域上的正规基是一个具有广泛理论和实际意义的课题。最近，$k$ -正规元素被引入作为正规元素的自然扩张。有限域的固定扩张中$k$ -正规元的存在性和个数问题是一个很有前途的研究方向。在本文中，我们首先制定了一个一般的下限的$k$ -正规元素的数目，假设它们存在。利用多项式$x^m-1$到分圆多项式的一般因式分解，我们进一步得到了$k$正规元的一个新的存在条件。最后，我们给出了$\fqm$中具有非极大但高乘法阶的正规元在有限域的单位群中存在的条件。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
