Related papers: Equidistribution of Primitive Normal Elements in F…

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^n-1$. We say that the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the line property for $r$-primitive elements property if, for every…

Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of a non-zero element $\epsilon \in…

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…

A primitive completely normal element for an extension $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ of Galois fields is a generator of the multiplicative group of $\mathbb{F}_{q^n}$, which simultaneously is normal over every intermediate field of that…

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we…

In this paper, the notion of a uniformly distributed systems of elements on the variety of metabelian Lie algebras is introduced. This notion is analogous to one of a measure preserving systems of elements on group varieties. As the main…

For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all…

A set of natural numbers $A$ is called primitive if no element of $A$ divides any other. Let $\Omega(n)$ be the number of prime divisors of $n$ counted with multiplicity. Let $f_z(A) = \sum_{a \in A}\frac{z^{\Omega(a)}}{a (\log a)^z}$,…

Let $\xi\in\mathbb{F}_{q^m}$ be an $r$-primitive $k$-normal element over $\mathbb{F}_q$, where $q$ is a prime power and $m$ is a positive integer. The minimal polynomial of $\xi$ is referred to be the $r$-primitive $k$-normal polynomial of…

This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the…

This is an essay about a certain family of elements in the general linear group GL(d,q) called primitive prime divisor elements, or ppd-elements. A classification of the subgroups of GL(d,q) which contain such elements is discussed, and the…

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 $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

Suppose $\ell$ is a prime number, ${\mathbf Q}_\ell$ is the field of $\ell$-adic numbers, ${\mathbf F}_\ell$ is the finite field of $\ell$ elements, and $d$ is a positive integer. Suppose $G$ is a finite subgroup of a symplectic group…

Let $\mathbb F_q$ be a finite field with $q$ elements, $G$ a finite cyclic group of order $p^k$ and $p$ is an odd prime with ${\rm gcd}(q,p)=1$. In this article, we determine an explicit expression for the primitive idempotents of $\mathbb…

Let $z\ne \pm1,w^2$ be a fixed integer, and let $f(t)\ne g(t)^2$ be a fixed polynomial over the integers. It is shown that the subset of primes $p\geq 2$ such that $z$ and $f(z)$ is a pair of simultaneous primitive roots modulo $p$ has…

In this paper we derive a formula for the number of $N$-free elements over a finite field $\mathbb{F}_q$ with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive a simple explicit formula…

We prove that for all $q>61$, every non-zero element in the finite field $\mathbb{F}_{q}$ can be written as a linear combination of two primitive roots of $\mathbb{F}_{q}$. This resolves a conjecture posed by Cohen and Mullen.

We introduce the notion of primitive elements in arbitrary truncated $p$-divisible groups. By design, the scheme of primitive elements is finite and locally free over the base. Primitive elements generalize the "points of exact order $N$,"…

Let $r > 0$ be an integer, let $\mathbb{F}_q$ be a finite field of $q$ elements, and let $\mathcal{A}$ be a nonempty proper subset of $\mathbb{F}_q$. Moreover, let $\mathbf{M}$ be a random $m \times n$ rank-$r$ matrix over $\mathbb{F}_q$…