# A structure theorem for finite fields

## TL;DR

This paper introduces a new structure theorem for finite fields of odd order, linking their multiplicative and additive properties, with applications to polynomials and number theory.

## Contribution

It presents a novel structure theorem for finite fields of odd order, enhancing understanding of their algebraic properties and applications.

## Key findings

- Improved understanding of Dickson and Chebyshev polynomials
- New formulas with number-theoretic significance
- Enhanced structural insights into finite fields

## Abstract

We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev polynomials and some formulas with a number-theoretic flavor. This paper is an abridged version of two math arXiv articles by the author [arXiv:1707.06870, arXiv:1707.06877].