This innovative undergraduate textbook approaches number theory through the lens of abstract algebra. Written in an engaging and whimsical style, this text will introduce students to rings, groups, fields, and other algebraic structures as they discover the key concepts of elementary number theory. Inquiry-based learning (IBL) appears throughout the chapters, allowing students to develop insights for upcoming sections while simultaneously strengthening their understanding of previously covered topics. The text is organized around three core themes: the notion of what a “number” is, and the premise that it takes familiarity with a large variety of number systems to fully explore number theory; the use of Diophantine equations as catalysts for introducing and developing structural ideas; and the role of abstract algebra in number theory, in particular the extent to which it provides the Fundamental Theorem of Arithmetic for various new number systems. Other aspects of modern number theory – including the study of elliptic curves, the analogs between integer and polynomial arithmetic, p-adic arithmetic, and relationships between the spectra of primes in various rings – are included in smaller but persistent threads woven through chapters and exercise sets.
Each chapter concludes with exercises organized in four categories: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness. IBL “Exploration” worksheets appear in many sections, some of which involve numerical investigations. To assist students who may not have experience with programming languages, Python worksheets are available on the book’s website. The final chapter provides five additional IBL explorations that reinforce and expand what students have learned, and can be used as starting points for independent projects. The topics covered in these explorations are public key cryptography, Lagrange’s four-square theorem, units and Pell’s Equation, various cases of the solution to Fermat’s Last Theorem, and a peek into other deeper mysteries of algebraic number theory.
Students should have a basic familiarity with complex numbers, matrix algebra, vector spaces, and proof techniques, as well as a spirit of adventure to explore the “numberverse.”
Author(s): Cam McLeman, Erin McNicholas, Colin Starr
Series: Undergraduate Texts in Mathematics
Publisher: Springer
Year: 2022
Language: English
Pages: 379
City: Cham
Preface
Who is This Text’s Audience?
To the Student:
To the Instructor:
Suggested Pacing and Content Coverage
Contents
1 What is a Number?
1.1 Human conception of numbers
1.2 Algebraic Number Systems
1.3 New Numbers, New Worlds
1.4 Exercises
2 A Quick Survey of the Last Two Millennia
2.1 Fermat, Wiles, and The Father of Algebra
2.2 Quadratic Equations
2.3 Diophantine Equations
2.4 Exercises
3 Number Theory in Z Beginning
3.1 Algebraic Structures
3.2 Linear Diophantine Equations and the Euclidean Algorithm
3.3 The Fundamental Theorem of Arithmetic
3.4 Factors and Factorials
3.5 The Prime Archipelago
3.6 Exercises
4 Number Theory in the Mod-n Era
4.1 Equivalence Relations and the Binary World
4.2 The Ring of Integers Modulo n
4.3 Reduce First and ask Questions Later
4.4 Division, Exponentiation, and Factorials in Zn
4.5 Group Theory and the Ring of Integers Modulo n
4.6 Lagrange's Theorem and the Euler Totient Function
4.7 Sunzi's Remainder Theorem and phi(n)
4.8 Phis, Polynomials, and Primitive Roots
4.9 Exercises
5 Gaussian Number Theory: Zi of the Storm
5.1 The Calm Before
5.2 Gaussian Divisibility
5.3 Gaussian Modular Arithmetic
5.4 Gaussian Division Algorithm: The Geometry of Numbers
5.5 A Gausso-Euclidean Algorithm
5.6 Gaussian Primes and Prime Factorizations
5.7 Applications to Diophantine Equations
5.8 Exercises
6 Number Theory, from Where We R to Across the C
6.1 From -1 to -d
6.2 Algebraic Numbers and Rings of Integers
6.3 Quadratic Fields: Integers, Norms, and Units
6.4 Euclidean Domains
6.5 Unique Factorization Domains
6.6 Euclidean Rings of Integers
6.7 Exercises
7 Cyclotomic Number Theory: Roots and Reciprocity
7.1 Introduction
7.2 Quadratic Residues and Legendre Symbols
7.3 Quadratic Residues and Non-Residues Mod p
7.4 Application: Counting Points on Curves
7.5 The Quadratic Reciprocity Law: Statement and Use
7.6 Some Unexpected Helpers: Roots of Unity
7.7 A Proof of Quadratic Reciprocity
7.8 Quadratic UFDs
7.9 Exercises
8 Number Theory Unleashed: Release Zp
8.1 The Analogy between Numbers and Polynomials
8.2 The p-adic World: An Analogy Extended
8.3 p-adic Arithmetic: Making a Ring
8.4 Which numbers are p-adic?
8.5 Hensel's Lemma
8.6 The Local-Global Philosophy and the Infinite Prime
8.7 The Local-Global Principle for Quadratic Equations
8.8 Computations: Quadratic Equations Made Easy
8.9 Synthesis and Beyond: Moving Between Worlds
8.10 Exercises
9 The Adventure Continues
9.1 Exploration: Fermat's Last Theorem for Small n
9.2 Exploration: Lagrange's Four-Square Theorem
9.3 Exploration: Public Key Cryptography
9.3.1 Public Key Encryption: RSA
9.3.2 Elliptic Curve Cryptography
9.3.3 Elliptic ElGamal Public Key Cryptosystem
9.4 Exploration: Units of Real Quadratic Fields
9.5 Exploration: Ideals and Ideal Numbers
9.6 Conclusion: The Numberverse, Redux
Appendix I Number Systems
I.1 Introduction
I.2 Construction of the Natural Numbers
I.3 Induction and Well-Ordering
Appendix Index
Index
Appendix Index of Notation
Author Index
