Number Theory 1: Fermat's Dream (Translations of Mathematical Monographs) (Vol 1)

This is the English translation of the original Japanese book. In this volume, "Fermat's Dream", core theories in modern number theory are introduced. Developments are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the number fields. This work presents an elegant perspective on the wonder of numbers. Number Theory 2 on class field theory, and Number Theory 3 on Iwasawa theory and the theory of modular forms, are forthcoming in the series. Cover Title page Contents Preface Preface to the English edition Objectives and Outline of these Books Notation Chapter 0. Introduction — Fermat and Number Theory — 0.1. Before Fermat 0.2. Prime numbers and the sum of two squares 0.3. p = x^2 + 2y^2, p = x^2 + 3y^2 , ... 0.4. Pell's equations 0.5. Triangular numbers, quadrangular numbers, pentagonal numbers 0.6. Triangular numbers, squares, cubes 0.7. Right triangles and elliptic curves 0.8. Fermat's Last Theorem Exercises Chapter 1. Rational Points on Elliptic Curves 1.1. Fermat and elliptic curves 1.2. Group structure of an elliptic curve 1.3. Mordell's theorem Summary Exercises Chapter 2. Conics and p-adic Numbers 2.1. Conics 2.2. Congruence 2.3. Conics and quadratic residue symbols 2.4. p-adic number fields 2.5. Multiplicative structure of the p-adic number field 2.6. Rational points on conics Summary Exercises Chapter 3. ζ 3.1. Three wonders of the values of the ζ function 3.2. Values at positive integers 3.3. Values at negative integers Summary Exercises Chapter 4. Algebraic Number Theory 4.1. Method of algebraic number theory 4.2. The heart of algebraic number theory 4.3. The class number formula for imaginary quadratic fields 4.4. Fermat's Last Theorem and Kummer Summary Exercises Appendix A. Rudiments on Dedekind domains A.1. Definition of a Dedekind domain A.2. Fractional ideal Answers to Questions Chapter 1 Chapter 2 Chapter 3 Chapter 4 Answers to Exercises Chapter 0 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Index Back Cover Introduction: Fermat And Number Theory. -- Rational Points On Elliptic Curves. -- Conics And P-adic Numbers. -- [zeta Function] -- Algebraic Number Theory. -- Appendix A. Rudiments On Dedekind Domains. Kazuya Kato, Nobushige Kurokawa, Takeshi Saito ; Translated By Masato Kuwata. Includes Index. Translated From The Japanese.