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

Arpan Chandra Mazumder, Giorgos Kapetanakis, Dhiren Kumar Basnet

Abstract

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 normal or primitive normal. In the former case, we fully characterize the conditions under which such elements exist and provide an explicit enumeration of these elements. In the latter case we provide asymptotic results.

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

Abstract

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 normal or primitive normal. In the former case, we fully characterize the conditions under which such elements exist and provide an explicit enumeration of these elements. In the latter case we provide asymptotic results.
Paper Structure (12 sections, 16 theorems, 34 equations)

This paper contains 12 sections, 16 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linearized polynomials and $\mathbb{F}_q$-order
  4. Characteristic functions
  5. The trace map
  6. Some estimates
  7. Intermediate Traces of Normal Elements
  8. Intermediate traces of primitive normal elements
  9. Existence results
  10. Remarks
  11. Acknowledgments
  12. Declarations

Key Result

Theorem 1.1

Let $q$ be a prime power, $m$ a positive integer and $a \in \mathbb{F}_q$. Then there exists a primitive element $\alpha \in \mathbb{F}_{q^m}$ such that $\mathop{\mathrm{Tr}}\nolimits_{m/1}(\alpha) = a$ unless $a = 0$ and $m = 2$ or $a = 0, m = 3$ and $q = 4$.

Theorems & Definitions (25)

  • Theorem 1.1: cohen1990
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Lemma 2.6: reis
  • Lemma 2.7: hucz2
  • Lemma 2.8: lens
  • Lemma 2.9
  • ...and 15 more