內容簡介

This book is intended to complement my Elements of Algebra, and it is similarly motivated by the problem of solving polynomial equations.However, it is independent of the algebra book, and probably easier. In Elements of Algebra we sought solution by radicals, and this led to theconcepts of fields and groups and their fusion in the celebrated theory of Galois. In the present book we seek integer solutions, and this leads to the concepts of rings and ideals which merge in the equally celebrated theo of ideals due to Kummer and Dedekind.
 

目錄

Preface
1 Natural numbers and integers
　1.1 Natural numbers
　1.2 Induction
　1.3 Integers
　1.4 Division with remainder
　1.5 Binary notation
　1.6 Diophantine equations
　1.7 TheDiophantus chord method
　1.8 Gaussian integers
　1.9 Discussion
2 The Euclidean algorithm
　2.1 The gcd by subtraction
　2.2 The gcd by division with remainder
　2.3 Linear representation of the gcd
　2.4 Primes and factorization
　2.5 Consequences of unique prime factorization
　2.6 Linear Diophantine equations
　2.7 *The vector Euclidean algorithm
　2.8 *The map of relatively prime pairs
　2.9 Discussion
3 Congruence arithmetic
　3.1 Congruence mod n
　3.2 Congruence classes and their arithmetic
　3.3 Inverses modp
3.4 Fermat’’s little theorem
　3.5 Congruence theorems of Wilson and Lagrange..
　3.6 Inversesmodk
　3.7 Quadratic Diophantine equations
　3.8 *Primitive roots
　3.9 *Existence of primitive roots
　3.10 Discussion
4 The RSA eryptosystem
4.1 Trapdoor functions
4.2 Ingredients of RSA
4.3 Exponentiation mod n
4.4 RSA encryption and decryption
4.5 Digital signatures
4.6 Other computational issues
4.7 Discussion
5 The Pell equation
5.1 Side and diagonal numbers
5.2 The equation x2 - 2y2 = 1
5.3 The group of solutions
5.4 The general Pell equation and
5.5 The pigeonhole argument
5.6 *Quadratic forms
5.7 *The map of primitive vectors
5.8 *Periodicity in the map ofx2 -ny2
5.9 Discussion
6 The Gaussian integers
6.1 Zand its norm
6.2 Divisibility and primes in Zand Z
6.3 Conjugates
6.4 Division in Z[i]
6.5 Fermat’’s two square theorem
6.6 Pythagorean triples
6.7 *Primes of the form 4n + 1
6.8 Discussion
……
7 Quadratic integers
8 The four square theorem
9 Quadratic reciprocity
10 Rings
11 Ideals
12 Prime ideals
Bibliography
Index
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