數論基礎（簡體書）

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.Solving equations in integers is the central problem of number theory,so this book is truly a number theory book, with most of the results found in standard number theory courses. However, numbers are best understood through their algebraic structure, and the necessary algebraic concepts--rings and ideals--have no better motivation than number theory.本書是一本講述數論的本科生教程。書中包括了一些有關的代數，並著重強調解決方程的整數解。尋找整數解引入數論的兩個最原始基本的問題―歐拉代數和素因子分解，及當代代數的基本靈魂―環和理想。這些思想的形成和從古代到現代的過度構成該書主要框架，歷史概念的講述引進新思想，現代證明的應用使得內容更加簡單、自然、有趣。書中包括一些從沒有以教材的形式出現的內容，如運用二次形式康威理論處理Pell方程。而且，這也是僅有的一本包括理想理論的數論書。敘述清楚，引用恰當，聯繫典型使得本書成為一本很難得教科書和自學課本。目次：自然數與整數；歐拉算法；同餘算術；密碼系統；Pell方程；Gaussian整數；二次整數；四次方理論；二次互反性；環；理想；素理想。
讀者對象：數學專業的本科生、研究生和有關專業的自學人員。
作者簡介：作者John Stillwell是San Francisco大學的教授。其作品有《數學及其歷史》，第2版，（2002），《數論和幾何》（1998），《代數基礎》（1994），這些著作品均已被Springer-Verlag出版。

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 Fermats 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 Fermats 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