arXiv 2007.111692020-07math.NT

有限域中本原$2$ -正规元的存在性

Existence of primitive $2$-normal elements in finite fields

Victor G. L. Neumann, Josimar J. R. Aguirre

摘要：

如果$\mathcal{B}=\{α, α^q, α^{q^2}, \cdots, α^{q^{n-1}}\}$作为$\mathbb{F}_q$上的向量空间形成$\mathbb{F}_{q^n}$的基，则元素$α\in \mathbb{F}_{q^n}$在$\mathbb{F}_q$上正规。众所周知，当且仅当$g_α(x)=αx^{n-1}+α^q x^{n-2}+ \cdots + α^{q^{n-2}}x+α^{q^{n-1}}$和$x^n-1$相对于$\mathbb{F}_{q^n}$互质，即它们在$\mathbb{F}_{q^n}[x]$中的最大公约数的次数是$0$时，$α\in \mathbb{F}_{q^n}$相对于$\mathbb{F}_q$是正态的。利用这个等价性，Huczynska等人（$2013$）引入了$k$ -正规元素的概念：如果$\mathbb{F}_{q^n}[x]$中多项式$g_α[x]$和$x^n-1$的最大公约数具有次数$k$，则元素$α\in \mathbb{F}_{q^n}$在$\mathbb{F}_q$上是$k$ -正规的;因此通常意义上正规的元素是$0$ -正规的。Huczynska等人提出了关于在$\mathbb{F}_q$上的$\mathbb{F}_{q^n}$中存在本原$k$ -正规元的$(n,k)$对的问题，并得到了$k=1$情形的部分结果，后来Reis和Thomson（$2018$）完成了这个情形。原始正规基定理解决了$k=0$的情况。本文利用Gauss和的估计及计算机的应用，完全解决了$k=2$情形，并得到了$\mathbb{F}_{q^n}$中$k$ -正规元存在的一个新条件.

英文摘要：

An element $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if $\mathcal{B}=\{α, α^q, α^{q^2}, \cdots, α^{q^{n-1}}\}$ forms a basis of $\mathbb{F}_{q^n}$ as a vector space over $\mathbb{F}_q$. It is well known that $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if and only if $g_α(x)=αx^{n-1}+α^q x^{n-2}+ \cdots + α^{q^{n-2}}x+α^{q^{n-1}}$ and $x^n-1$ are relatively prime over $\mathbb{F}_{q^n}$, that is, the degree of their greatest common divisor in $\mathbb{F}_{q^n}[x]$ is $0$. Using this equivalence, the notion of $k$-normal elements was introduced in Huczynska et al. ($2013$): an element $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α[x]$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$; so an element which is normal in the usual sense is $0$-normal. Huczynska et al. made the question about the pairs $(n,k)$ for which there exist primitive $k$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ and they got a partial result for the case $k=1$, and later Reis and Thomson ($2018$) completed this case. The Primitive Normal Basis Theorem solves the case $k=0$. In this paper, we solve completely the case $k=2$ using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of $k$-normal elements in $\mathbb{F}_{q^n}$.

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