Topology of Numbers

This book serves as an introduction to number theory at the undergraduate level, emphasizing geometric aspects of the subject. The geometric approach is exploited to explore in some depth the classical topic of quadratic forms with integer coefficients, a central topic of the book. Quadratic forms of this type in two variables have a very rich theory, developed mostly by Euler, Lagrange, Legendre, and Gauss during the period 1750–1800. In this book their approach is modernized by using the splendid visualization tool introduced by John Conway in the 1990s called the topograph of a quadratic form. Besides the intrinsic interest of quadratic forms, this theory has also served as a stepping stone for many later developments in algebra and number theory. The book is accessible to students with a basic knowledge of linear algebra and arithmetic modulo $n$. Some exposure to mathematical proofs will also be helpful. The early chapters focus on examples rather than general theorems, but theorems and their proofs play a larger role as the book progresses. Front Cover Contents Preface Chapter 0. A Preview Chapter 1. The Farey Diagram 1.1. The Mediant Rule 1.2. Farey Series Chapter 2. Continued Fractions 2.1. Finite Continued Fractions 2.2. Infinite Continued Fractions 2.3. Linear Diophantine Equations Chapter 3. Symmetries of the Farey Diagram 3.1. Linear Fractional Transformations 3.2. Translations and Glide Reflections Chapter 4. Quadratic Forms 4.1. The Topograph 4.2. Periodicity 4.3. Pell’s Equation Chapter 5. Classification of Quadratic Forms 5.1. The Four Types of Forms 5.2. Equivalence of Forms 5.3. The Class Number 5.4. Symmetries of Forms 5.5. Charting All Forms Chapter 6. Representations by Quadratic Forms 6.1. Three Levels of Complexity 6.2. Representations in a Fixed Discriminant 6.3. Genus and Characters 6.4. Proof of Quadratic Reciprocity Chapter 7. The Class Group for Quadratic Forms 7.1. Multiplication of Forms 7.2. The Class Group for Forms 7.3. Finite Abelian Groups 7.4. Symmetry and the Class Group 7.5. Genus and the Class Group 7.6. Rational Equivalence and Rational Forms Chapter 8. Quadratic Fields 8.1. Prime Factorization 8.2. Unique Factorization via the Euclidean Algorithm 8.3. The Correspondence Between Forms and Ideals 8.4. The Ideal Class Group 8.5. Unique Factorization of Ideals 8.6. Applications to Forms Tables 1. Forms of Negative Discriminant 2. Fully Symmetric Negative Discriminants 3. Forms of Positive Nonsquare Discriminant 4. Periodic Separator Lines Nonstandard Terminology Bibliography Index Provides an introduction to number theory at the undergraduate level, emphasizing geometric aspects of the subject. The geometric approach is exploited to explore in depth the classical topic of quadratic forms with integer coefficients, a central topic of the book.