Elements of Number Theory

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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.
 

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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 y x + 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 x + y x + 2y and x + 3y
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 x + 5y
233
129 Discussion
236
Bibliography
239
Index
245
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