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Arithmetic progression in a finite field with prescribed norms

Kaustav Chatterjee, Hariom Sharma, Aastha Shukla, Shailesh Kumar Tiwari

Abstract

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 progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for $n\geq6,q=3^k,m=2$ we establish that there are only $10$ possible exceptions.

Arithmetic progression in a finite field with prescribed norms

Abstract

Given a prime power and a positive integer , let represents a finite extension of degree of the finite field . In this article, we investigate the existence of elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for we establish that there are only possible exceptions.
Paper Structure (9 sections, 17 theorems, 53 equations, 4 tables)

This paper contains 9 sections, 17 theorems, 53 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Some useful results
  5. Main Results
  6. Definitions.
  7. Working example
  8. Part I
  9. Part II

Key Result

Lemma 3.1

(LDQ, Theorem 5.5) Let $f(x)=\prod_{j=1}^{s}f_{j}(x)^{a_{j}}$ be a rational function over $\mathbb{F}_{q^{n}}$, where $f_{j}(x)\in\mathbb{F}_{q^{n}}[x]$ are polynomials and $a_{j}\in\mathbb{Z}\smallsetminus\{0\}$. Let $\chi\in\widehat{{\mathbb{F}}_{q}^*}$ be a multiplicative character of order $d$.

Theorems & Definitions (24)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • ...and 14 more