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

Maithri K., Vadiraja Bhatta G. R., Indira K. P., Prasanna Poojary

TL;DR

This work extends order-theoretic concepts from finite-field elements to polynomials by introducing the ${\mathbb F}_q$-Order of a polynomial and exploring its structural parallels with element orders. Using this framework, it analyzes $r$-primitive and $k$-normal elements and derives exact counting formulas for irreducible polynomials of a given degree with fixed ${\mathbb F}_q$-Orders, as well as for $k$-normal polynomials. It also establishes how these orders behave under field extensions, products, and sums, and investigates trace properties linked to the ${\mathbb F}_q$-Order, providing a unified, combinatorial perspective that connects element-based and polynomial-based order theories with applications to finite-field cryptography and coding theory.

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.

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

TL;DR

This work extends order-theoretic concepts from finite-field elements to polynomials by introducing the -Order of a polynomial and exploring its structural parallels with element orders. Using this framework, it analyzes -primitive and -normal elements and derives exact counting formulas for irreducible polynomials of a given degree with fixed -Orders, as well as for -normal polynomials. It also establishes how these orders behave under field extensions, products, and sums, and investigates trace properties linked to the -Order, providing a unified, combinatorial perspective that connects element-based and polynomial-based order theories with applications to finite-field cryptography and coding theory.

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 -Order for elements have been extensively studied. In this paper, we investigate several properties of -primitive and -normal elements. Furthermore, by using the concept of the -Order of a polynomial, we explore properties of -normal polynomials.
Paper Structure (8 sections, 25 theorems, 22 equations)

This paper contains 8 sections, 25 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiplicative order and $r$-primitive elements
  4. $\mathbb{F}_q$-Order and $k$-normal elements
  5. $m$-free and $g$-free elements
  6. $\mathbb{F}_q$-Order of a Polynomial
  7. Algebraic and Combinatorial Properties of $r$-Primitive and $k$-Normal Elements
  8. Conclusions

Key Result

Proposition 4

If $f \in \mathbb{F}_q[x]$ is irreducible polynomial over $\mathbb{F}_q$ of degree $m$, then $Ord(f)=Ord(\alpha)$, where $\alpha$ is the root of $f$.

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Example 3
  • Proposition 4
  • proof
  • Corollary 5
  • Proposition 6
  • proof
  • Corollary 7
  • Proposition 8
  • ...and 37 more