Topics In Normal Bases Of Finite Fields

TL;DR

This paper provides a comprehensive introduction to the theory of normal bases in finite fields, covering standard results, special types like Gaussian and period bases, and asymptotic existence proofs for primitive polynomials.

Contribution

It offers a detailed overview of normal bases, including proofs and new asymptotic results on primitive polynomials with specific properties.

Findings

Existence of primitive polynomials with about half arbitrary coefficients.

Asymptotic proofs for primitive normal polynomials of arbitrary trace.

In-depth coverage of Gaussian and period normal bases.

Abstract

This is an introduction to the theory of normal bases of finite fields. The first few chapters cover a wide range of topics on the theory of normal bases of finite fields. Most standard definitions and results, including proofs are given. The last few chapters cover the theory of guassian and period normal bases of finite fields of low degrees. The last chapter presents the asymptotic proofs of the existence of primitive polynomials of degree n with approximately n/2 arbitrary coefficients, and primitive normal polynomials of arbitrary traces.