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Small Primitive Normal Elements in Finite Fields

N. A. Carella

TL;DR

The paper proves that every nonstructured subset $\mathcal{A}$ of $\mathbb{F}_{q^n}$ with cardinality $\#\mathcal{A}\gg q^{\varepsilon}$ contains a primitive normal element as $q^n\to\infty$, and provides a quantified lower bound $N_q(\mathcal{A})\gg (\log\log q^n)^{\varepsilon}$. Central to the result is the development of divisor-free indicator functions for primitive and normal elements that avoid factoring $q^n-1$ and $x^n-1$, paired with sharp estimates for incomplete character sums and exponential sums. The authors integrate two analytic frameworks— novel characteristic functions and precise sum bounds—to establish existence in small, nonstructured sets and to derive effective counting asymptotics. The work yields a practical benchmark for deterministic search complexity of primitive normal elements and opens several density-related questions, including the distribution of primitive normal elements in the algebraic closure of the base field. Overall, the paper advances the understanding of primitive normal elements in finite fields and provides robust tools for their detection in small subsets.

Abstract

Let $q=p^k$ be a prime power, let $\mathbb{F}_q$ be a finite field and let $n\geq2$ be an integer. This note investigates the existence small primitive normal elements in finite field extensions $\mathbb{F}_{q^n}$. It is shown that a small nonstructured subset $\mathcal{A}\subset \mathbb{F}_{q^n}$ of cardinality $\#\mathcal{A}\gg q^{\varepsilon}$, where $\varepsilon>0$ is a small number, contains a primitive normal element.

Small Primitive Normal Elements in Finite Fields

TL;DR

The paper proves that every nonstructured subset of with cardinality contains a primitive normal element as , and provides a quantified lower bound . Central to the result is the development of divisor-free indicator functions for primitive and normal elements that avoid factoring and , paired with sharp estimates for incomplete character sums and exponential sums. The authors integrate two analytic frameworks— novel characteristic functions and precise sum bounds—to establish existence in small, nonstructured sets and to derive effective counting asymptotics. The work yields a practical benchmark for deterministic search complexity of primitive normal elements and opens several density-related questions, including the distribution of primitive normal elements in the algebraic closure of the base field. Overall, the paper advances the understanding of primitive normal elements in finite fields and provides robust tools for their detection in small subsets.

Abstract

Let be a prime power, let be a finite field and let be an integer. This note investigates the existence small primitive normal elements in finite field extensions . It is shown that a small nonstructured subset of cardinality , where is a small number, contains a primitive normal element.
Paper Structure (39 sections, 39 theorems, 162 equations)

This paper contains 39 sections, 39 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Notation, Definitions and Basic Concepts
  3. Small Elements and Subsets
  4. Norm and Trace Functions
  5. Linearized Polynomials
  6. Additive Characters and Additive Orders
  7. Multiplicative Characters and Multiplicative Orders
  8. Normal Element Test
  9. Primitive Element Test
  10. Fundamental Identities
  11. Estimates for Character Sums
  12. Estimates for Incomplete Character Sums over Finite Rings
  13. Estimates for Incomplete Character Sums over Finite Fields
  14. Estimates of Exponential Sums over Subgroups
  15. Average Orders of Mobius Functions
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $q=p^k$ be a prime power and let $\varepsilon>0$ be a small number. Let $\mathcal{A}\subset \mathbb{F}_{q^n}$ be a nonstructured subset of cardinality Then, as $q^n\to \infty$, the subset $\mathcal{A}$ contains primitive normal elements. In particular, for large $q^n> q^2$, the weighted counting function for the number of primitive normal elements has the asymptotic lower bound

Theorems & Definitions (83)

  • Theorem 1.1
  • Corollary 1.1
  • Conjecture 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 73 more