Related papers: $r$-primitive $k$-normal elements in arithmetic pr…

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 $\alpha \in \mathbb{F}_{q^n}^*$ is called $r$-primitive if its multiplicative order is $(q^n-1)/r$. Also, $\alpha \in…

Let $r$, $n$ be positive integers, $k$ be a non-negative integer and $q$ be any prime power such that $r\mid q^n-1.$ An element $\alpha$ of the finite field $\mathbb{F}_{q^n}$ is called an {\it $r$-primitive} element, if its multiplicative…

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of $k$-normal elements: an element $\alpha \in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_{\alpha}(x)=…

An element $\alpha \in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if $\mathcal{B}=\{\alpha, \alpha^q, \alpha^{q^2}, \cdots, \alpha^{q^{n-1}}\}$ forms a basis of $\mathbb{F}_{q^n}$ as a vector space over $\mathbb{F}_q$. It is well known…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. An element $\alpha \in \mathbb{F}_{q^n}$ is called $k$-normal over $\mathbb{F}_q$ if $\alpha$ and its conjugates generate a vector subspace of $\mathbb{F}_{q^n}$ of dimension…

An element $\alpha \in \mathbb F_{q^n}$ is \emph{normal} if $\mathcal{B} = \{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ forms a basis of $\mathbb F_{q^n}$ as a vector space over $\mathbb F_{q}$; in this case, $\mathcal{B}$ is a normal…

For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive…

An element $\alpha \in \mathbb {F}_{q^n}$ is normal over $\mathbb {F}_q$ if $\alpha$ and its conjugates $\alpha, \alpha^q, \cdots \alpha^{q^{n-1}}$ form a basis of $\mathbb {F}_{q^n}$ over $\mathbb {F}_q$. Recently, Huczynska, Mullen,…

Let $\mathbb{F}_{q^n}$ be the extension of the field $\mathbb{F}_q$ of degree n, where $q$ is power of prime $p$, i.e $q=p^k$, where k is a positive integer. In this paper, we provide sufficient condition for the existence of a primitive…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the…

Polynomials and elements over finite fields exhibit closely related algebraic structures, and many properties defined for elements extend naturally to polynomials. The concepts of order and $\mathbb{F}_q$-Order for elements have been…

Given $\mathbb{F}_{q^{n}}$, a field with $q^n$ elements, where $q $ is a prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2 \in \mathbb{N}$, $k_1,k_2 \in \mathbb{N}\cup \{0\}$, a rational function $F = \frac{F_1}{F_2}$ in…

Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ represents a finite extension of degree $n$ of the finite field ${\mathbb{F}_{q}}$. In this article, we investigate the existence of $m$ elements in arithmetic…

Let $q$ be a prime power of a prime $p$, $n$ a positive integer and $\mathbb F_{q^n}$ the finite field with $q^n$ elements. The $k-$normal elements over finite fields were introduced and characterized by Huczynska et al (2013). Under the…

In this paper, we explore the existence of $m$-terms arithmetic progressions in $\mathbb{F}_{q^n}$ with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for…

For each positive integer $n$, let $\mathbb F_{q^n}$ be the unique $n$-degree extension of the finite field $\mathbb F_q$ with $q$ elements, where $q$ is a prime power. It is known that for arbitrary $q$ and $n$, there exists an element…

The so called $k$-normal elements appear in the literature as a generalization of normal elements over finite fields. Recently, questions concerning the construction of $k$-normal elements and the existence of $k$-normal elements that are…

The celebrated Primitive Normal Basis Theorem states that for any $n\ge 2$ and any finite field $\mathbb F_q$, there exists an element $\alpha\in \mathbb F_{q^n}$ that is simultaneously primitive and normal over $\mathbb F_q$. In this…

Let $q$ be a prime power and, for each positive integer $n\ge 1$, let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements. Motivated by the well known concept of normal elements over finite fields, Huczynska et al (2013) introduced…

An element $\alpha \in \mathbb{F}_{q^n}$ is a normal element over $\mathbb{F}_q$ if the conjugates $\alpha^{q^i}$, $0 \leq i \leq n-1$, are linearly independent over $\mathbb{F}_q$. Hence a normal basis for $\mathbb{F}_{q^n}$ over…