数论基础_PDF下载[133MB-百度云]史迪威

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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.

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Preface1 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 Discussion2 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 Discussion4 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 Discussion5 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 Discussion6 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 integers8 The four square theorem9 Quadratic reciprocity10 Rings11 Ideals12 Prime idealsBibliographyIndex