What question did this study set out to answer?

To investigate the existence of primitive and normal elements in finite fields of odd characteristic.

March 29, 2026

A special type of primitive normal pair in a finite field of odd characteristic

KCKaustav ChatterjeeOklahoma State University STS. K. TiwariIndian Institute of Technology Patna HSHariom SharmaGovernment of Haryana

Key Points

To investigate the existence of primitive and normal elements in finite fields of odd characteristic.
Examined the properties of finite field extensions of degree n.
Analyzed conditions for elements to be primitive and normal simultaneously.
Considered specific odd prime powers and calculated exceptions for element choices.
Identified guaranteed existence of primitive normal pairs for most cases when n is specified.
Noted exact 3 exceptions where primitive normal pairs do not exist.

Abstract

Let Formula: see text with Formula: see text an odd prime power, and let Formula: see text denote the degree-Formula: see text extension of the finite field Formula: see text. In this paper, we study the existence of an element Formula: see text for which both Formula: see text and Formula: see text are simultaneously primitive and normal over Formula: see text. Furthermore, when Formula: see text for some Formula: see text, such a pair is guaranteed except for exact 3 choices of Formula: see text.

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