Table Of Content

Universitext Forothertitlesinthisseries,goto www.springer.com/series/223 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 (cid:176)©c SpringerScience(cid:0)+ BusinessMedia,LLC2009 Allrightsreserved. Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentiﬁedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper Springer is part of Springer Science+Business Media (www.springer.com) 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