Related papers: Existence of primitive $1$-normal elements in fini…

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…

Denote by $\mathbb F_q$ the finite field of order $q$ and by $\mathbb F_{q^n}$ its extension of degree $n$. Some $a\in\mathbb F_{q^n}$ is called primitive if it generates the multiplicative group $\mathbb F_{q^n}^*$ and it is called…

Let $q=p^k$ be a prime power, let $\mathbb{F}_q$ be a finite field and let $n\geq2$ be an integer. This note investigates the existence small primitive normal elements in finite field extensions $\mathbb{F}_{q^n}$. It is shown that a small…

In this paper we generalize the results of Sharma, Awasthi and Gupta (see \cite{SAG}). We work over a field of any characteristic with $q = p^k$ elements and we give a sufficient condition for the existence of a primitive element $\alpha…

In this paper, we consider rational functions $f$ with some minor restrictions over the finite field $\mathbb{F}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a…

With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $\gamma$ generates $\Fqn$ over $\Fq$ and $\beta$ is a non-zero element of $\Fqn$, is there always an $a \in \Fq$ such that $\beta(\gamma + a)$ is a…

Let $q, n, m \in \mathbb{N}$ such that $q$ is a prime power, $m \geq 3$ and $a \in \mathbb{F}$. We establish a sufficient condition for the existence of a primitive normal pair ($\alpha$, $f(\alpha)$) in $\mathbb{F}_{q^m}$ over…

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…

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 $m, n, q\in \mathbb{N}$ such that $q$ is a prime power and $m\geq 3$, $a\in \mathbb{F}_q$, we establish a sufficient condition for the existence of primitive pair $(\alpha, f(\alpha))$ in $\mathbb{F}_{q^m}$ such that $\alpha$ is…

In this paper we study elements $\beta \in \mathbb{F}_{q^n}$ having normal $\alpha$-depth $b$; that is, elements for which $\beta, \beta - \alpha, \ldots, \beta-(b-1)\alpha$ are simultaneously normal elements of $\mathbb{F}_{q^n}$ over…

An element w of the extension E of degree n over the finite field F=GF(q) is called free over F if {w, w^q,...,w^{q^{n-1}}} is a (normal) basis of E/F. The Primitive Normal Basis Theorem, first established in full by Lenstra and Schoof…

In this article, we establish a sufficient condition for the existence of primitive element $\alpha\in \Fm$ is such that $f(\alpha)$ is also primitive element of $\Fm$ and $Tr_{\Fm/\F}(\alpha)=\beta$, for any prescribed $\beta\in\F$, where…

Let $q, n, m \in \mathbb{N}$ be such that $q$ is a prime power and $a, b \in \mathbb{F}$. In this article we establish a sufficient condition for the existence of a primitive normal pair $(\alpha, f(\alpha)) \in \mathbb{F}_{q^m}$ over…

Let $\mathbb{F}_q$ be the finite field of $q$ elements, and let $k\mid q-1$ be a positive integer. Let $f(x)=ax^2+bx+c$ be a quadratic polynomial in $\mathbb{F}_q[x]$ with $b^2-4ac\ne0$. In this paper, we show that if…

Let $q=p^k$ be a prime power, let $n\geq2$ be an integer and let $\mathbb{F}_{q^n}$ be a finite field. It is shown that the set of primitive normal elements is a Salem set. Furthermore, it is proved that this set is strongly equidistributed…

Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary members of the ground field $\mathbb{F}_{q}$. We investigate the existence of a non-zero…

This article investigates the existence of an $r$-primitive $k$-normal polynomial, defined as the minimal polynomial of an $r$-primitive $k$-normal element in $\mathbb{F}_{q^n}$, with a specified degree $n$ and two given coefficients over…

The notion of normal elements for finite fields extension has been generalized as k-normal elements by Huczynska et al. [3]. The number of k-normal elements for a fixed finite field extension has been calculated and estimated [3], and…

For a finite field $\mathbf{F}_{q^r}$ with fixed $q$ and $r$ sufficiently large, we prove the existence of a primitive element outside of a set of $r$ many affine hyperplanes for $q=4$ and $q=5$. This complements earlier results by…