G. Kapetanakis

activity

20172026

most citedFinite field extensions with the line or translate property for

r

-primitive elements

3 citations · 4 across the 17 of their papers we have counts for

collaborators

20 papers

math.NT2026

An estimate for incomplete mixed character sums and applications

Arpan Chandra Mazumder, Giorgos Kapetanakis, Sushant Kala +1

Let $q$ be a prime power and $m > 1$ be any integer. Let $F_{q^{m}}$ be the finite field of order $q^{m}$ and $θ \in F_{q^{m}}$ be such that $F_{q^{m}} = F (θ)$ .…

math.NT2025

Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields

Arpan Chandra Mazumder, Giorgos Kapetanakis, Dhiren Kumar Basnet

In this article, we study the existence and distribution of elements in finite field extensions with prescribed traces in several intermediate extensions that are also either norma…

math.NT2025★ 1 cited

Translates of completely normal elements and the Morgan-Mullen conjecture

Theodoulos Garefalakis, Giorgos Kapetanakis

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

math.NT2024

Existence of Special Types Primitive Pairs in Finite Fields Avoiding Affine Hyperplanes

Himangshu Hazarika, Giorgos Kapetanakis, Dhiren Kumar Basnet

Let $\Fm$ be finite fields of order $q^{m}$ , where $m \geq 2$ and $q$ , a prime power. Given $\F$-affine hyperplanes $A_{1}, \dots, A_{m}$ of $\Fm$ in general position, we study the existe…

math.NT2024

Normal points on Artin-Schreier curves over finite fields

Giorgos Kapetanakis, Lucas Reis

In 2022, S.D. Cohen and the two authors introduced and studied the concept of $(r, n)$ -freeness on finite cyclic groups $G$ for suitable integers $r, n$ , which is an arithmetic way…

math.NT2024

$F_{q}$ -primitive points on varieties over finite fields

Soniya Takshak, Giorgos Kapetanakis, Rajendra Kumar Sharma

Let $r$ be a positive divisor of $q - 1$ and $f (x, y)$ a rational function of degree sum $d$ over $F_{q}$ with some restrictions, where the degree sum of a rational function $f…

math.NT2024

Existence of primitive normal pairs over finite fields with prescribed subtrace

K. Chatterjee, G. Kapetanakis, H. Sharma +1

Given positive integers $q, n, m$ and $a \in F_{q}$ , where $q$ is an odd prime power and $n \geq 5$ , we investigate the existence of a primitive normal pair $(ε, f (ε))$ in $\mat…

math.HO2023

Teaching perspectives of the Frobenius coin problem of two denominators

Giorgos Kapetanakis, Ioannis Rizos

Let $a, b$ be positive, relatively prime, integers. Our goal is to characterize, in an elementary way, all positive integers $c$ that can be expressed as a linear combination of $a,…

math.NT2023

An inductive proof of the Frobenius coin problem of two denominations

Giorgos Kapetanakis, Ioannis Rizos

Let $a, b$ be positive, relatively prime, integers. We prove, using induction, that for every $d > ab - a - b$ there exist $x, y \in Z_{\geq 0}$ , such that $d = a x + b y$ . As a byprodu…

math.NT2023

Primitive normal pairs of elements with one prescribed trace

Arpan Chandra Mazumder, Himangshu Hazarika, Dhiren Kumar Basnet +1

Let $q, n, m \in N$ such that $q$ is a prime power, $m \geq 3$ and $a \in F$ . We establish a sufficient condition for the existence of a primitive normal pair ($α…

math.NT2021

The trace of primitive and $2$ -primitive elements in finite fields, revisited

Stephen D. Cohen, Giorgos Kapetanakis

By definition primitive and $2$ -primitive elements of a finite field extension $F_{q^{n}}$ have order $q^{n} - 1$ and $(q^{n} - 1) /2$ , respectively. We have already shown that, with…

math.NT2021

The existence of $F_{q}$ -primitive points on curves using freeness

Stephen D. Cohen, Giorgos Kapetanakis, Lucas Reis

Let $C_{Q}$ be the cyclic group of order $Q$ , $n$ a divisor of $Q$ and $r$ a divisor of $Q / n$ . We introduce the set of $(r, n)$ -free elements of $C_{Q}$ and derive a l…

math.NT2020

On existence of primitive normal elements of rational form over finite fields of even characteristic

Himangshu Hazarika, Dhiren Kumar Basnet, Giorgos Kapetanakis

Let $q$ be an even prime power and $m \geq 2$ an integer. By $F_{q}$ , we denote the finite field of order $q$ and by $F_{q^{m}}$ its extension degree $m$ . In this paper…

math.NT2019

The translate and line properties for 2-primitive elements in quadratic extensions

Stephen D. Cohen, Giorgos Kapetanakis

Let $r, n > 1$ be integers and $q$ be any prime power $q$ such that $r ∣ q^{n} - 1$ . We say that the extension $F_{q^{n}} / F_{q}$ possesses the line property for $r$ -primit…

math.NT2019★ 3 cited

Finite field extensions with the line or translate property for $r$ -primitive elements

Stephen D. Cohen, Giorgos Kapetanakis

Let $r, n > 1$ be integers and $q$ be any prime power $q$ such that $r ∣ q^{n} - 1$ . We say that the extension $F_{q^{n}} / F_{q}$ possesses the line property for $r$ -primit…

math.NT2019

The trace of 2-primitive elements of finite fields (amended version)

Stephen D. Cohen, Giorgos Kapetanakis

Let $q$ be a prime power and $n, r$ integers such that $r ∣ q^{n} - 1$ . An element of $F_{q^{n}}$ of multiplicative order $(q^{n} - 1) / r$ is called \emph{ $r$ -primitive}. For any o…

math.NT2018

A Swan-like note for a family of binary pentanomials

Giorgos Kapetanakis

In this note, we employ the techniques of Swan (Pacific J. Math. 12(3): 1099-1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the pen…

math.NT2018

Further results on the Morgan-Mullen conjecture

Theodoulos Garefalakis, Giorgos Kapetanakis

Let $F_{q}$ be the finite field of characteristic $p$ with $q$ elements and $F_{q^{n}}$ its extension of degree $n$ . The conjecture of Morgan and Mullen asserts the e…

math.NT2017

Variations of the Primitive Normal Basis Theorem

Giorgos Kapetanakis, Lucas Reis

The celebrated Primitive Normal Basis Theorem states that for any $n \geq 2$ and any finite field $F_{q}$ , there exists an element $α \in F_{q^{n}}$ that is simultaneously…

math.NT2017

On the existence of primitive completely normal bases of finite fields

Theodoulos Garefalakis, Giorgos Kapetanakis

Let $F_{q}$ be the finite field of characteristic $p$ with $q$ elements and $F_{q^{n}}$ its extension of degree $n$ . We prove that there exists a primitive element of…

An estimate for incomplete mixed character sums and applications

Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields

Translates of completely normal elements and the Morgan-Mullen conjecture

Existence of Special Types Primitive Pairs in Finite Fields Avoiding Affine Hyperplanes

Normal points on Artin-Schreier curves over finite fields

Fq​-primitive points on varieties over finite fields

Existence of primitive normal pairs over finite fields with prescribed subtrace

Teaching perspectives of the Frobenius coin problem of two denominators

An inductive proof of the Frobenius coin problem of two denominations

Primitive normal pairs of elements with one prescribed trace

The trace of primitive and 2-primitive elements in finite fields, revisited

The existence of Fq​-primitive points on curves using freeness

On existence of primitive normal elements of rational form over finite fields of even characteristic

The translate and line properties for 2-primitive elements in quadratic extensions

Finite field extensions with the line or translate property for r-primitive elements

The trace of 2-primitive elements of finite fields (amended version)

A Swan-like note for a family of binary pentanomials

Further results on the Morgan-Mullen conjecture

Variations of the Primitive Normal Basis Theorem

On the existence of primitive completely normal bases of finite fields

$F_{q}$ -primitive points on varieties over finite fields

The trace of primitive and $2$ -primitive elements in finite fields, revisited

The existence of $F_{q}$ -primitive points on curves using freeness

Finite field extensions with the line or translate property for $r$ -primitive elements