Number Theory 1,9780821808634

Number Theory 1

by ; ; ;
ISBN13: 9780821808634
ISBN10: 082180863X
Format: Paperback
Pub. Date: 2000-01-01
Publisher(s): Amer Mathematical Society
List Price: $39.59

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Summary

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.

Table of Contents

Preface ix
Preface to the English Edition xi
Objectives and Outline of these Books xiii
Notation xv
Introduction --- Fermat and Number Theory 1(1)
Before Fermat
1(3)
Prime numbers and the sum of two squares
4(2)
p = x2 + 2y2, p = x2 + 3y2,...
6(1)
Pell's equations
7(1)
Triangular numbers, quadrangular numbers, pentagonal numbers
8(2)
Triangular numbers, squares, cubes
10(1)
Right triangles and elliptic curves
11(1)
Fermat's Last Theorem
12(5)
Exercises
14(3)
Rational Points on Elliptic Curves
17(28)
Fermat and elliptic curves
17(8)
Group structure of an elliptic curve
25(5)
Mordell's theorem
30(15)
Summary
43(1)
Exercises
43(2)
Conics and p-adic Numbers
45(36)
Conics
45(4)
Congruence
49(4)
Conics and quadratic residue symbols
53(5)
p-adic number fields
58(11)
Multiplicative structure of the p-adic number field
69(5)
Rational points on conics
74(7)
Summary
78(1)
Exercises
78(3)
ζ
81(22)
Three wonders of the values of the ζ function
81(3)
Values at positive integers
84(5)
Values at negative integers
89(14)
Summary
99(1)
Exercises
100(3)
Algebraic Number Theory
103(32)
Method of algebraic number theory
104(9)
The heart of algebraic number theory
113(11)
The class number formula for imaginary quadratic fields
124(3)
Fermat's Last Theorem and Kummer
127(8)
Summary
132(1)
Exercises
132(3)
Appendix A. Rudiments on Dedekind domains 135(4)
A.1. Definition of a Dedekind domain
135(1)
A.2. Fractional ideal
136(3)
Answers to Questions 139(6)
Answers to Exercises 145(8)
Index 153

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