Generators of finite fields with prescribed traces

TL;DR

This paper investigates the existence of primitive elements in finite fields with specific trace conditions across intermediate extensions, providing new inequalities and concrete existence results.

Contribution

It introduces a novel inequality condition guaranteeing primitive elements with prescribed traces and applies it to particular intermediate extensions.

Findings

Established an inequality condition for existence of primitive elements with prescribed traces.

Derived concrete existence results for specific classes of intermediate extensions.

Enhanced understanding of primitive element distribution in finite fields.

Abstract

This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.