Table Of Content
Universitext
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W.A. Coppel
Number Theory
An Introduction to Mathematics
Second Edition
W.A. Coppel
3 Jansz Crescent
2603 Griffith
Australia
Editorialboard:
SheldonAxler,SanFranciscoStateUniversity
VincenzoCapasso,UniversitàdegliStudidiMilano
CarlesCasacuberta,UniversitatdeBarcelona
AngusMacIntyre,QueenMary,UniversityofLondon
KennethRibet,UniversityofCalifornia,Berkeley
ClaudeSabbah,CNRS,ÉcolePolytechnique
EndreSüli,UniversityofOxford
WojborWoyczyn´ski,CaseWesternReserveUniversity
ISBN 978-0-387-89485-0 e-IS BN 978-0 -387-89486-7
DOI 10 .1007/978-0-387-89486-7
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2009931687
Mathematics Subject Classification (2000): 11-xx, 05B20, 33E05
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ForJonathan,Nicholas,PhilipandStephen
Contents
PrefacetotheSecondEdition ....................................... xi
PartA
I TheExpandingUniverseofNumbers............................ 1
0 Sets,RelationsandMappings ................................. 1
1 NaturalNumbers............................................ 5
2 IntegersandRationalNumbers ................................ 10
3 RealNumbers .............................................. 17
4 MetricSpaces .............................................. 27
5 ComplexNumbers .......................................... 39
6 QuaternionsandOctonions ................................... 48
7 Groups .................................................... 55
8 RingsandFields ............................................ 60
9 VectorSpacesandAssociativeAlgebras ........................ 64
10 InnerProductSpaces ........................................ 71
11 FurtherRemarks ............................................ 75
12 SelectedReferences ......................................... 79
AdditionalReferences ............................................ 82
II Divisibility .................................................. 83
1 GreatestCommonDivisors ................................... 83
2 TheBe´zoutIdentity.......................................... 90
3 Polynomials................................................ 96
4 EuclideanDomains.......................................... 104
5 Congruences ............................................... 106
6 SumsofSquares ............................................ 119
7 FurtherRemarks ............................................ 123
8 SelectedReferences ......................................... 126
AdditionalReferences ............................................ 127
viii Contents
III MoreonDivisibility .......................................... 129
1 TheLawofQuadraticReciprocity ............................. 129
2 QuadraticFields ............................................ 140
3 MultiplicativeFunctions...................................... 152
4 LinearDiophantineEquations................................. 161
5 FurtherRemarks ............................................ 174
6 SelectedReferences ......................................... 176
AdditionalReferences ............................................ 178
IV ContinuedFractionsandTheirUses ............................ 179
1 TheContinuedFractionAlgorithm............................. 179
2 DiophantineApproximation................................... 185
3 PeriodicContinuedFractions.................................. 191
4 QuadraticDiophantineEquations .............................. 195
5 TheModularGroup ......................................... 201
6 Non-EuclideanGeometry..................................... 208
7 Complements............................................... 211
8 FurtherRemarks ............................................ 217
9 SelectedReferences ......................................... 220
AdditionalReferences ............................................ 222
V Hadamard’sDeterminantProblem ............................. 223
1 WhatisaDeterminant? ...................................... 223
2 HadamardMatrices.......................................... 229
3 TheArtofWeighing......................................... 233
4 SomeMatrixTheory......................................... 237
5 ApplicationtoHadamard’sDeterminantProblem................. 243
6 Designs.................................................... 247
7 GroupsandCodes........................................... 251
8 FurtherRemarks ............................................ 256
9 SelectedReferences ......................................... 258
VI Hensel’sp-adicNumbers ...................................... 261
1 ValuedFields............................................... 261
2 Equivalence ................................................ 265
3 Completions................................................ 268
4 Non-ArchimedeanValuedFields............................... 273
5 Hensel’sLemma ............................................ 277
6 LocallyCompactValuedFields................................ 284
7 FurtherRemarks ............................................ 290
8 SelectedReferences ......................................... 290
Contents ix
PartB
VII TheArithmeticofQuadraticForms............................. 291
1 QuadraticSpaces............................................ 291
2 TheHilbertSymbol ......................................... 303
3 TheHasse–MinkowskiTheorem............................... 312
4 Supplements ............................................... 322
5 FurtherRemarks ............................................ 324
6 SelectedReferences ......................................... 325
VIII TheGeometryofNumbers .................................... 327
1 Minkowski’sLatticePointTheorem............................ 327
2 Lattices.................................................... 330
3 ProofoftheLatticePointTheorem;OtherResults ................ 334
4 VoronoiCells............................................... 342
5 DensestPackings............................................ 347
6 Mahler’sCompactnessTheorem............................... 352
7 FurtherRemarks ............................................ 357
8 SelectedReferences ......................................... 360
AdditionalReferences ............................................ 362
IX TheNumberofPrimeNumbers ................................ 363
1 FindingtheProblem ......................................... 363
2 Chebyshev’sFunctions....................................... 367
3 ProofofthePrimeNumberTheorem ........................... 370
4 TheRiemannHypothesis..................................... 377
5 GeneralizationsandAnalogues................................ 384
6 AlternativeFormulations ..................................... 389
7 SomeFurtherProblems ...................................... 392
8 FurtherRemarks ............................................ 394
9 SelectedReferences ......................................... 395
AdditionalReferences ............................................ 398
X ACharacterStudy ........................................... 399
1 PrimesinArithmeticProgressions ............................. 399
2 CharactersofFiniteAbelianGroups............................ 400
3 ProofofthePrimeNumberTheoremforArithmeticProgressions ... 403
4 RepresentationsofArbitraryFiniteGroups ...................... 410
5 CharactersofArbitraryFiniteGroups .......................... 414
6 InducedRepresentationsandExamples ......................... 419
7 Applications................................................ 425
8 Generalizations ............................................. 432
9 FurtherRemarks ............................................ 443
10 SelectedReferences ......................................... 444
