CONTENTS
Preface = xi
Readers' guide = xv
Index of notation = xvii
The origins of algebraic number theory = 1
PART Ⅰ ALGEBRAIC METHODS
1 Algebraic background = 9
1.1 Rings and fields = 10
1.2 Factorization of polynomials = 14
1.3 Field extensions = 22
1.4 Symmetric polynomials = 24
1.5 Modules = 27
1.6 Free abelian groups = 28
2 Algebraic numbers = 38
2.1 Algebraic numbers 39
2.2 Conjugates and discriminants = 41
2.3 Algebraic integers = 45
2.4 Integral bases = 50
2.5 Norms and traces = 54
2.6 Rings of integers = 56
3 Quadratic and cyclotomic fields = 66
3.1 Quadratic fields = 66
3.2 Cyclotomic fields = 69
4 Factorization into irreducibles = 78
4.1 Historical background = 79
4.2 Trivial factorizations = 81
4.3 Factorization into irreducibles = 85
4.4 Examples of non-unique factorization into irreducibles = 89
4.5 Prime factorization = 94
4.6 Euclidean domains = 97
4.7 Euclidean quadratic fields = 99
4.8 Consequences of unique factorization = 102
4.9 The Ramanujan-Nagell theorem = 104
5 Ideals = 110
5.1 Historical background = 111
5.2 Prime factorization of ideals = 114
5.3 The norm of an ideal = 125
5.4 Nonunique factorization in cyclotomic fields = 133
PART Ⅱ GEOMETRIC METHODS
6 Lattices = 141
6.1 Lattices = 141
6.2 The quotient torus = 144
7 Minkowski's theorem = 151
7.1 Minkowski's theorem = 151
7.2 The two-squares theorem = 154
7.3 The four-squares theorem = 155
8 Geometric representation of algebraic numbers = 158
8.1 The space Lst = 158
9 Class-group and class-number = 164
9.1 The class-group = 164
9.2 An existence theorem = 166
9.3 Finiteness of the class-group = 171
9.4 How to make an ideal principal = 172
9.5 Unique factorization of elements in an extension ring = 176
PART Ⅲ NUMBER-THEORETIC APPLICATIONS
10 Computational methods = 185
10.1 Factorization of a rational prime = 185
10.2 Minkowski's constants = 189
10.3 Some class-number calculations = 193
10.4 Tables = 196
11 Format's Last Theorem = 199
11.1 Some history = 199
11.2 Elementary considerations = 202
11.3 Kummer's lemma = 205
11.4 Kummer's Theorem = 210
11.5 Regular primes = 214
12 Dirichlet's Units Theorem = 219
12.1 Introduction = 219
12.2 Logarithmic space = 220
12.3 Embedding the unit group in logarithmic space = 221
12.4 Dirichlet's theorem = 223
Appendix Quadratic Residues = 231
A-l Quadratic equations in Zm = 232
A.2 The units of Zm = 234
A.3 Quadratic Residues = 240
References = 253
Index = 257
