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Primitive elements of finite fields $\mathbf{F}_{q^r}$ avoiding affine hyperplanes for $q=4$ and $q=5$

Philipp Alexander Grzywaczyk, Arne Winterhof

Abstract

For a finite field $\mathbf{F}_{q^r}$ with fixed $q$ and $r$ sufficiently large, we prove the existence of a primitive element outside of a set of $r$ many affine hyperplanes for $q=4$ and $q=5$. This complements earlier results by Fernandes and Reis for $q\ge 7$. For $q=3$ the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For $q=2$ the set consists only of a single element, and such a result is thus not possible.

Primitive elements of finite fields $\mathbf{F}_{q^r}$ avoiding affine hyperplanes for $q=4$ and $q=5$

Abstract

For a finite field with fixed and sufficiently large, we prove the existence of a primitive element outside of a set of many affine hyperplanes for and . This complements earlier results by Fernandes and Reis for . For the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For the set consists only of a single element, and such a result is thus not possible.
Paper Structure (3 sections, 8 theorems, 31 equations)

This paper contains 3 sections, 8 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results for Primitive Elements Avoiding Affine Hyperplanes

Key Result

Lemma 2.1

For $U\subseteq \mathds{F}_q^\ast$, the number $\mathcal{P}(U)$ of primitive elements in $U$ is given by where $\varphi$ denotes Euler's totient function, $\mu$ the Möbius-function and $\widehat{\mathds{F}}_q^\ast$ the group of multiplicative characters of $\mathds{F}_q$.

Theorems & Definitions (12)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 2 more