Elements of Number Theory

Springer Science & Business Media, Dec 13, 2002 - Mathematics - 256 pages
This book is intended to complement my Elements oi 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 oi Algebra we sought solution by radicals, and this led to the concepts 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 theory 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. The first nontrivial examples of rings appear in the number theory of Euler and Gauss. The concept of ideal-today as routine in ring the ory as the concept of normal subgroup is in group theory-also emerged from number theory, and in quite heroic fashion. Faced with failure of unique prime factorization in the arithmetic of certain generalized "inte gers" , Kummer created in the 1840s a new kind of number to overcome the difficulty. He called them "ideal numbers" because he did not know exactly what they were, though he knew how they behaved.

What people are saying -Write a review

User Review - Flag as inappropriate

Number theory is probably the oldest branch of mathematics; as Stillwell points out, one of the earliest pieces of human writing that we possess is a four thousand year old clay tablet with a list of what we would now call Pythagorean triples. Number theory also occupies a strange place in modern mathematical education. High schools don't teach number theory, and I am an example of the fact that it's possible to graduate with a degree in math from a good university without ever taking a single class in the subject. Until recently I didn't realize that it was still an active field of research; I thought of it as just a collection of amusing trivia. Silly me. I've picked up some elementary facts about number theory more or less by osmosis (everyone with a well rounded technical education has at least heard of Euclid's proof of the infinitude of primes, or modulo arithmetic, or Fermat's little theorem), but I've never read a discussion of how number theory forms a coherent whole. What I found most interesting about this book was the connection between number theory and abstract algebra. I learned about ideals in my abstract algebra classes, and I thought of them as more or less the equivalent of normal subgroups for rings. Now that I've read this book I finally understand why ideals are interesting. It's a good illustration of how you can understand abstractions better after you've seen the concrete examples that the abstractions came from.

Contents

 Natural numbers and integers 1 11 Natural numbers 2 12 Induction 3 13 Integers 5 14 Division with remainder 7 15 Binary notation 8 16 Diophantine equations 11 17 The Diophantus chord method 14
 72 The division property in Z2 119 73 The gcd in Zv2 121 74 Z3andZC3 123 75 Rational solutions of jc3 + y3 z3 + w3 126 76 The prime y3 in Zf3 129 77 Fermats last theorem for n 3 132 78 Discussion 136 The four square theorem 138

 18 Gaussian integers 17 19 Discussion 20 The Euclidean algorithm 22 22 The gcd by division with remainder 24 23 Linear representation of the gcd 26 24 Primes and factorization 28 25 Consequences of unique prime factorization 30 26 Linear Diophantine equations 33 27 The vector Euclidean algorithm 35 28 The map of relatively prime pairs 38 29 Discussion 40 Congruence arithmetic 43 31 Congruence mod n 44 32 Congruence classes and their arithmetic 45 33 Inverses mod p 48 34 Fermats little theorem 51 35 Congruence theorems of Wilson and Lagrange 53 36 Inverses mod k 55 37 Quadratic Diophantine equations 57 38 Primitive roots 59 39 Existence of primitive roots 62 310 Discussion 63 The RSA cryptosystem 66 42 Ingredients of RSA 69 43 Exponentiation mod n 70 44 RSA encryption and decryption 72 45 Digital signatures 73 46 Other computational issues 74 The Pell equation 76 51 Side and diagonal numbers 77 52 The equation x2 2y2 l 78 53 The group of solutions 80 54 The general Pell equation and Zyn 81 55 The pigeonhole argument 84 56 Quadratic forms 87 57 The map of primitive vectors 90 58 Periodicity in the map of x2 ny2 95 59 Discussion 99 The Gaussian integers 101 61 Zi and its norm 102 62 Divisibility and primes in Zi and Z 103 63 Conjugates 105 64 Division in Zi 107 65 Fermats two square theorem 109 66 Pythagorean triples 110 67 Primes of the form 4n + 1 113 68 Discussion 115 Quadratic integers 117 71 The equation y³ x² + 2 118
 81 Real matrices and C 139 82 Complex matrices and H 141 83 The quaternion units 143 84 Zijk 145 85 The Hurwitz integers 147 86 Conjugates 149 87 A prime divisor property 151 88 Proof of the four square theorem 152 89 Discussion 154 Quadratic reciprocity 158 91 Primes x² + y² x² + 2y² and x² + 3y² 159 92 Statement of quadratic reciprocity 161 93 Eulers criterion 164 94 The value of 167 95 The story so far 169 96 The Chinese remainder theorem 171 97 The full Chinese remainder theorem 173 98 Proof of quadratic reciprocity 175 99 Discussion 178 Rings 181 101 The ring axioms 182 102 Rings and fields 184 103 Algebraic integers 186 104 Quadratic fields and their integers 189 105 Norm and units of quadratic fields 192 106 Discussion 194 Ideals 196 111 Ideals and the gcd 197 112 Ideals and divisibility in Z 199 199 113 Principal ideal domains 202 114 A nonprincipal ideal of Zv3 205 115 A nonprincipal ideal of Zv5 207 116 Ideals of imaginary quadratic fields as lattices 209 117 Products and prime ideals 211 118 Ideal prime factorization 214 119 Discussion 217 Prime ideals 221 121 Ideals and congruence 222 122 Prime and maximal ideals 224 123 Prime ideals of imaginary quadratic fields 225 124 Conjugate ideals 227 125 Divisibility and containment 229 126 Factorization of ideals 230 127 Ideal classes 231 128 Primes of the form x² + 5y² 233 129 Discussion 236 Bibliography 239 Index 245 Copyright