论文标题
$ r $ $ - 主要和$ k $ - 正常元素在有限领域
Pairs of $r$-primitive and $k$-normal elements in finite fields
论文作者
论文摘要
令$ \ mathbb {f} _ {q^n} $为有限字段，带有$ q^n $ elements，$ r $为$ q^n-1 $的正分数。如果其乘法订单为$（q^n-1）/r $，则属于\ mathbb {f} _ {q^n}^*$的元素$α\。另外，$α\ in \ mathbb {f} _ {q^n} $是$ k $ - normal of $ \ mathbb {f} _q $如果多项式$g_α（x）=αx^{n-1} +α^q x x x x x x x x^x^^n-2} + c $g_α（x）=g_α（x） α^{q^{n-2}} x +α^{q^{n-1}} $和$ x^n-1 $ in $ \ mathbb {f} _ {q^n} [q^n} [x] $具有$ k $。这些概念分别概括了原始元素和正常元素的思想。在本文中，我们考虑非阴性整数$ m_1，m_2，k_1，k_2 $，正整数$ r_1，r_2 $和有理函数$ f（x）= f_1（x）/f_2（x）/f_2（x）\ in \ in \ in \ mathbb {f} $ i \ in \ {1,2 \} $满足某些条件，我们为存在$ r_1 $ - primive $ k_1 $ -normal元素$α\ in \ mathbb {f} _ {q^n} $超过了$ \ m m i \ im，例如$ k_2 $ - 正态元素超过$ \ m athbb {f} _q $。最后，作为一个例子，我们研究了$ r_1 = 2 $，$ r_2 = 3 $，$ k_1 = 2 $，$ k_2 = 1 $，$ m_1 = 2 $和$ m_2 = 1 $，带有$ n \ ge 7 $。
Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements and $r$ be a positive divisor of $q^n-1$. An element $α\in \mathbb{F}_{q^n}^*$ is called $r$-primitive if its multiplicative order is $(q^n-1)/r$. Also, $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α(x) = αx^{n-1}+ α^q x^{n-2} + \ldots + α^{q^{n-2}}x + α^{q^{n-1}}$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integers $m_1,m_2,k_1,k_2$, positive integers $r_1,r_2$ and rational functions $F(x)=F_1(x)/F_2(x) \in \mathbb{F}_{q^n}(x)$ with $°(F_i) \leq m_i$ for $i\in\{ 1,2\}$ satisfying certain conditions and we present sufficient conditions for the existence of $r_1$-primitive $k_1$-normal elements $α\in \mathbb{F}_{q^n}$ over $\mathbb{F}_q$, such that $F(α)$ is an $r_2$-primitive $k_2$-normal element over $\mathbb{F}_q$. Finally as an example we study the case where $r_1=2$, $r_2=3$, $k_1=2$, $k_2=1$, $m_1=2$ and $m_2=1$, with $n \ge 7$.
