From natural numbers to prime fields and finite fields

TL;DR

This book provides an accessible introduction to algebra by exploring the construction and properties of natural numbers, rational numbers, prime fields, and finite fields, emphasizing their foundational significance.

Contribution

It offers a self-contained presentation of constructing prime and finite fields, including proofs of their existence and uniqueness, with minimal prerequisites.

Findings

Constructs all prime fields and finite fields from basic set theory.

Proves the existence and uniqueness of finite fields.

Provides an accessible, self-contained exposition of algebraic field constructions.

Abstract

The aim of this book is to introduce the reader to the beauty of Algebra, through a journey from the natural numbers to prime fields and finite fields, with some detours. Many books are devoted to the construction of these fields from the natural numbers. Perhaps the first one is [Landau 1930], see also the books [Ebbinghaus et.al. 1991], [Lay 2005], and the references therein. An important step in this process is the construction of the field of rational numbers. It turns out that this field is the only (in an appropriate sense) infinite prime field. Motivated by the reading of [Loonstra 1972], I decided to write an essentially self-contained book devoted to the construction not only of the field of rational numbers but of all prime fields and, more interestingly, to the proof of existence and "uniqueness" of all finite fields. Only a knowledge of basic set theory and some familiarity with mathematical reasoning (see for example [11, Chapters 1 and 2]) are assumed.