| Topology
of Numbers |
This is an introduction to elementary number theory from a geometric point of view, as opposed to the usual strictly algebraic approach. A large part of the book is devoted to studying quadratic forms in two variables with integer coefficients, a very classical topic going back to Fermat, Euler, Lagrange, Legendre, and Gauss, but from a perspective that emphasizes Conway's much more recent notion of the topograph of a quadratic form.
You can download a pdf file of the current version of the book, around 320 pages, posted in December 2020. The book is now essentially complete and I am trying to do finial polishing. A few more exercises need to be added, especially in the later chapters. Publication should be in the first half of 2021.
Chapter 0. Preview
Chapter 1. The Farey Diagram
The Mediant Rule.
Farey Series.
Chapter 2. Continued Fractions
The Euclidean Algorithm.
Linear Diophantine Equations
Infinite Continued Fractions.
Chapter 3. Symmetries of the Farey Diagram
Linear Fractional Transformations.
Continued Fractions Again.
Chapter 4. Quadratic Forms
The Topograph.
Periodic Separator Lines.
Continued Fractions Once More.
Pell's Equation.
Chapter 5. Classification of Quadratic Forms
The Four Types of Forms.
Equivalence of Forms.
The Class Number.
Symmetries of Forms.
Charting All Forms.
Chapter 6. Representations by Quadratic Forms
Three Levels of Complexity.
Representions in a Fixed Discriminant.
Genus and Characters.
Proof of Quadratic Reciprocity.
Chapter 7. The Class Group for Quadratic Forms
Multiplication of Forms.
The Class Group for Forms.
Finite Abelian Groups.
Symmetry and the Class Group.
Genus and the Class Group.
Chapter 8. Quadratic Fields
Prime Factorization.
Unique Factorization via the Euclidean Algorithm.
The Correspondence Between Forms and Ideals.
The Ideal Class Group.
Unique Factorization of Ideals.
Applications to Forms.