Diophantine equations and power integral bases theory

Foreword......Page 6
Preface......Page 7
Acknowledgments......Page 11
Contents......Page 12
Notation......Page 18
 1.1 Some Basics of Number Fields......Page 20
 1.2 The Index of Algebraic Integers......Page 22
 1.3 Relative Extensions......Page 25
 1.4 Factors of the Index in Composite Fields......Page 28
 1.5 Results on the Field Index......Page 30
 2.1 Baker's Method and Effective Finiteness Theorems......Page 31
  2.2.1 Davenport Lemma......Page 33
  2.2.2 The General Case......Page 35
 2.3 Enumeration Methods......Page 36
 2.4 Software and Hardware......Page 41
 2.5 Related Results......Page 42
 3.1 Elementary Estimates......Page 43
 3.3 Fast Algorithm for Finding ``Small'' Solutions......Page 44
  3.4.1 Results on the Solutions of Binomial Thue Equations......Page 45
  3.4.2 Calculation of ``Small'' Solutions of Binomial Thue Equations......Page 47
 3.5 Effective Methods for Thue Equations......Page 50
 3.6 The Method of Bilu and Hanrot......Page 52
 3.7 Experiences by Solving a Thue Equation with ``Large'' Entries......Page 53
 3.8 Further Results on Thue Equations......Page 55
 4.1 Elementary Estimates......Page 56
 4.2 Baker's Method......Page 58
 4.3 Reduction, Test......Page 59
 4.4 An Analogue of the Bilu-Hanrot Method......Page 60
 5.1 A Fast Algorithm for Finding ``Small'' Solutions of Relative Thue Equations......Page 62
  5.1.1 The Reduction Procedure......Page 65
  5.1.2 Enumerating Tiny Values of the Variables......Page 67
  5.1.4 Computational Aspects......Page 69
  5.1.5 Examples......Page 70
  5.2.1 Specialties of the Actual Calculations......Page 73
  5.3.1 Baker's Method and Reduction......Page 74
  5.3.2 Enumeration......Page 77
  5.3.3 An Example......Page 78
 5.4 Totally Real Thue Equations over Imaginary Quadratic Fields......Page 80
  5.4.1 How to Apply Theorem 5.9......Page 86
  5.4.2 An Example......Page 87
 5.5 Simplest Quartic and Simplest Sextic Thue Equations over Imaginary Quadratic Fields......Page 88
  5.5.1 Proof of Theorem 5.11......Page 90
 5.6 Further Results on Relative Thue Equations......Page 96
 6.1 Preliminaries......Page 97
 6.2 Solving the Unit Equation......Page 99
 6.3 Calculating the Solutions of the Norm Form Equation......Page 101
 6.4 Examples......Page 102
 6.5 Further Results on Norm Form Equations......Page 108
 7.1 The Structure of the Index Form......Page 109
 7.2 Using Resolvents......Page 111
 7.5 Composite Fields......Page 112
  7.5.1 Coprime Discriminants......Page 113
  7.5.2 Non-coprime Discriminants 1......Page 115
  7.5.3 Non-coprime Discriminants 2......Page 117
 7.6 Notes on Montes Algorithm......Page 119
 8.1 Arbitrary Cubic Fields......Page 120
 8.2 Simplest Cubic Fields......Page 122
  8.3.1 Integral Basis: Index Form......Page 123
  8.3.2 Frequency of Monogenic Fields......Page 124
 8.4 Further Results on the Monogenity of Cubic Fields......Page 126
 9.1 Algorithm for Arbitrary Quartic Fields......Page 127
  9.1.1 The Resolvent Equation......Page 128
  9.1.2 The Quartic Thue Equations......Page 129
  9.1.3 Proof of Theorem 9.6......Page 132
  9.1.4 Examples......Page 137
  9.2.1 Power Integral Bases and Minimal Indices......Page 138
  9.2.2 Proof of Theorem 9.16......Page 141
 9.4 Totally Complex Quartic Fields......Page 146
  9.4.1 Parametric Families of Totally Complex Quartic Fields......Page 148
  9.4.2 A Family of Totally Complex Quartic Fields with Two Parameters......Page 150
  9.5.1 Integral Basis and Index Form......Page 153
  9.5.2 The Totally Real Case......Page 155
  9.5.3 The Totally Complex Case......Page 157
  9.5.4 The Field Index of Bicyclic Biquadratic Number Fields......Page 158
 9.6 Pure Quartic Fields......Page 161
 9.7 Further Results on the Monogenity of Quartic Fields......Page 164
 10.1 Algorithm for Arbitrary Quintic Fields......Page 165
  10.1.1 Preliminaries......Page 166
  10.1.2 Baker's Method and Reduction......Page 167
  10.1.3 Enumeration......Page 169
  10.1.4 Examples......Page 170
 10.2 Lehmer's Quintics......Page 175
  10.2.1 Integer Basis and Unit Group......Page 176
  10.2.2 The Index Form......Page 178
  10.2.3 The Index Form Equation......Page 180
  10.2.4 The Exceptional Case......Page 182
  11.1.1 The Unit Equation......Page 183
  11.1.2 Enumeration......Page 185
  11.1.3 An Example......Page 188
 11.2 Sextic Fields with a Quadratic Subfield......Page 190
  11.2.1 Real Quadratic Subfield......Page 192
  11.2.2 Totally Real Sextic Fields with a Quadratic and a Cubic Subfield......Page 193
  11.2.4 Sextic Fields with an Imaginary Quadratic and a Real Cubic Subfield......Page 194
  11.2.5 Parametric Families of Sextic Fields with Imaginary Quadratic and Real Cubic Subfields......Page 197
 11.3 Sextic Fields with a Cubic Subfield......Page 201
  11.4.2 A Noncyclic Sextic Field......Page 203
  11.5.1 Monogenity of an Order of ZKt......Page 204
  11.5.3 Monogenity of Simplest Sextic Fields......Page 205
  11.5.4 Computational Remarks......Page 208
 11.6 Further Results on the Monogenity of Sextic Fields......Page 209
 12.1 Integral Basis of Pure Fields......Page 210
 12.2 Monogenity of Pure Fields......Page 211
  12.2.1 Pure Cubic Fields: K=Q([3]m)......Page 212
  12.2.2 Pure Quartic Fields: K=Q([4]m)......Page 213
  12.2.3 Pure Sextic Fields: K=Q([6]m)......Page 214
  12.2.4 Pure Octic Fields: K=Q([8]m)......Page 216
 12.3 Notes......Page 217
 13.1 The Cubic Relative Thue Equation......Page 219
  13.1.1 Example 1: Cubic Extension of a Quintic Field......Page 221
  13.1.2 Example 2: Cubic Extension of a Sextic Field......Page 223
  13.1.3 Computational Experiences......Page 226
 13.2 Nonic Fields with Cubic Subfields......Page 227
  13.2.1 The Relative Thue Equations......Page 228
  13.2.2 The Unit Equation over the Normal Closure......Page 229
  13.2.3 The Common Variables......Page 231
 13.3 Composites of Cubic Fields with Real Quadratic Fields......Page 235
  13.3.1 Example......Page 237
  13.3.2 Computational Aspects......Page 239
  14.1.1 Preliminaries......Page 240
  14.1.2 The Cubic Relative Thue Equation......Page 241
  14.1.3 Representing the Variables as Binary Quadratic Forms......Page 243
  14.1.5 An Example: Computing Relative Power Integral Bases in a Quartic Extension of a Cubic Subfield......Page 245
  14.2.1 Preliminaries......Page 248
  14.2.2 The Unit Equation......Page 249
  14.2.3 The Inhomogeneous Thue Equation......Page 251
  14.2.4 Sieving......Page 252
  14.2.5 An Example......Page 253
 14.3 Composites of Quartic Fields with Real Quadratic Fields......Page 256
  14.3.1 Composites of Totally Complex Quartic Fields with Real Quadratic Fields......Page 258
  14.3.2 An Example......Page 260
 14.4 Pure Quartic Extensions of Imaginary Quadratic Fields......Page 263
  14.4.1 From Index Equations to Binomial Thue Equations......Page 264
  14.4.2 Generators of Relative Power Integral Bases......Page 265
  14.4.3 Absolute Power Integral Bases of Pure Quartic Extensions of Imaginary Quadratic Fields......Page 266
  14.5.1 Example 1: A Galois Family......Page 267
  14.5.2 Example 2: Simplest D4 Octics......Page 271
  14.5.3 Example 3: An Infinite Family with Quadratic Algebraic Parameter......Page 273
 15.1 Power Integral Bases in Imaginary Quadratic Extensions of Totally Real Cyclic Fields of Prime Degree......Page 276
 15.2 Power Integral Bases in Imaginary Quadratic Extensions of Lehmer's Quintics......Page 279
 15.4 Further Results on the Monogenity of Higher Degree Number Fields......Page 280
 15.5 Notes on the Monogenity of Cyclotomic Fields......Page 281
 16.1 Binomial Thue Equations......Page 283
 16.2 Binomial Thue Equations over Imaginary Quadratic Fields......Page 290
 16.3 Cubic Fields......Page 292
  16.3.1 Totally Real Cubic Fields......Page 293
  16.3.2 Complex Cubic Fields......Page 295
 16.4 Pure Cubic Fields......Page 296
  16.5.1 The Distribution of the Minimal Indices......Page 298
  16.5.2 The Average Behavior of the Minimal Indices......Page 299
  16.5.3 Totally Real Cyclic Quartic Fields......Page 300
  16.5.4 Monogenic Mixed Dihedral Extensions of Real Quadratic Fields......Page 301
  16.5.5 Totally Real Bicyclic Biquadratic Number Fields......Page 302
  16.5.6 Totally Complex Bicyclic Biquadratic Number Fields......Page 307
   16.5.7.1 Totally Real Quartic Fields with Galois Group A4......Page 309
   16.5.7.2 Totally Real Quartic Fields with Galois Group S4......Page 310
   16.5.7.3 Quartic Fields of Mixed Signature......Page 312
   16.5.7.4 Totally Complex Quartic Fields with Galois Group A4......Page 313
  16.6.1 Totally Real Cyclic Sextic Fields......Page 314
  16.6.2 Sextic Fields with Imaginary Quadratic Subfields......Page 316
 16.7 Integral Basis of the Simplest Sextic Fields......Page 319
References......Page 322
Index......Page 332