On \(\mathbb{F}_q\)-Order of Polynomials and Properties of \(r\)-Primitive and \(k\)-Normal Elements over Finite Fields

TL;DR

This paper explores the algebraic properties of polynomials and elements over finite fields, focusing on \\mathbb{F}_q-Order, r-primitive, and k-normal elements, and introduces new insights into k-normal polynomials.

Contribution

It introduces the concept of \\mathbb{F}_q-Order of polynomials and studies their relation to r-primitive and k-normal elements over finite fields.

Findings

Characterization of r-primitive elements using \\mathbb{F}_q-Order

Properties of k-normal polynomials derived from polynomial \\mathbb{F}_q-Order

New criteria for identifying k-normal polynomials

Abstract

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 extensively studied. In this paper, we investigate several properties of $r$-primitive and $k$-normal elements. Furthermore, by using the concept of the $\mathbb{F}_q$-Order of a polynomial, we explore properties of $k$-normal polynomials.