作者：史迪威
页数：254
出版社：世界图书出版公司
出版日期：2009
ISBN：9787510004674
电子书格式：pdf/epub/txt
内容简介
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.
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.
目录
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 最The vector Euclidean algorithm
2.8 最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 最Primitive roots
3.9 最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 最Quadratic forms
5.7 最The map of primitive vectors
5.8 最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 最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
