关于高斯周期
On Gauss Periods
摘要：
令$q$是素数幂，且令$r=nk+1$是素数使得$r\nmid q$，其中$n$和$k$是正整数。在$q$、$r$和$k$上的一个简单条件下，$(n,k)$型的高斯周期是$\Bbb F_q$上的$\Bbb F_{q^n}$的正规元;由此得到的$\Bbb F_q$上的$\Bbb F_{q^n}$的正规基的复杂性用$C(n,k;q)$表示。最近的工作确定了$k\le 7$和所有合格的$n$和$q$的$C(n,k;q)$。本文证明了对任何给定的$k>0$，除了有许多素数$r=nk+1$外，$C(n,k;q)$都由一个显式公式给出，并且例外素数很容易确定.此外，我们描述了一个算法，允许一个计算$C(n,k;q)$的例外素数$r=nk+1$。本文的数值结果涵盖了$k\le 20$的$C(n,k;q)$和所有合格的$n$和$q$。
英文摘要：
Let $q$ be a prime power, and let $r=nk+1$ be a prime such that $r\nmid q$, where $n$ and $k$ are positive integers. Under a simple condition on $q$, $r$ and $k$, a Gauss period of type $(n,k)$ is a normal element of $\Bbb F_{q^n}$ over $\Bbb F_q$; the complexity of the resulting normal basis of $\Bbb F_{q^n}$ over $\Bbb F_q$ is denoted by $C(n,k;q)$. Recent works determined $C(n,k;q)$ for $k\le 7$ and all qualified $n$ and $q$. In this paper, we show that for any given $k>0$, $C(n,k;q)$ is given by an explicit formula except for finitely many primes $r=nk+1$ and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute $C(n,k;q)$ for the exceptional primes $r=nk+1$. The numerical results of the paper cover $C(n,k;q)$ for $k\le 20$ and all qualified $n$ and $q$.
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。
