AlgebraicNumberTheoryforBeginners Thisbookintroducesalgebraicnumbertheorythroughtheproblemofgeneralizing “uniqueprimefactorization”fromordinaryintegerstomoregeneraldomains.Solving polynomialequationsinintegersleadsnaturallytothesedomains,butuniqueprime factorizationmaybelostintheprocess.Torestoreit,weneedDedekind’sconceptof ideals.However,onestillneedsthesupportingconceptsofalgebraicnumberfieldand algebraicinteger,andthesupportingtheoryofrings,vectorspaces,andmodules.It waslefttoEmmyNoethertoencapsulatethepropertiesofringsthatmakeunique primefactorizationpossible,inwhatwenowcallDedekindrings.Thebookdevelops thetheoryoftheseconcepts,followingtheirhistory,motivatingeachconceptualstep bypointingtoitsorigins,andfocusingonthegoalofuniqueprimefactorizationwitha minimumofdistractionorprerequisites.Thismakesforaself-contained,easy-to-read book,shortenoughforaone-semestercourse. john stillwellistheauthorofmanybooksonmathematics;amongthebest knownareMathematicsandItsHistory,NaiveLieTheory,andElementsof Mathematics.HeisamemberoftheinauguralclassofFellowsoftheAmerican MathematicalSocietyandwinneroftheChauvenetPrizeformathematicalexposition. Published online by Cambridge University Press Published online by Cambridge University Press Algebraic Number Theory for Beginners Following a Path from Euclid to Noether JOHN STILLWELL UniversityofSanFrancisco Published online by Cambridge University Press UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi 110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781316518953 DOI:10.1017/9781009004138 ©JohnStillwell2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-316-51895-3Hardback ISBN978-1-009-00192-2Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Published online by Cambridge University Press Tomygrandchildren,IdaandIsaac Published online by Cambridge University Press Published online by Cambridge University Press Contents Preface pagexi Acknowledgments xiv 1 EuclideanArithmetic 1 1.1 DivisorsandPrimes 2 1.2 TheFormofthegcd 5 1.3 ThePrimeDivisorProperty 8 1.4 IrrationalNumbers 10 1.5 TheEquationx2−2y2 =1 13 1.6 Rings 15 1.7 Fields 19 1.8 FactorsofPolynomials 22 1.9 Discussion 24 2 DiophantineArithmetic 33 2.1 RationalversusIntegerSolutions 34 2.2 Fermat’sLastTheoremforFourthPowers 36 2.3 SumsofTwoSquares 38 2.4 GaussianIntegersandPrimes 41 2.5 UniqueGaussianPrimeFactorization 43 2.6 FactorizationofSumsofTwoSquares 45 2.7 GaussianPrimes 47 2.8 PrimesthatAreSumsofTwoSquares 48 2.9 TheEquationy3 =x2+2 50 2.10 Discussion 53 3 QuadraticForms 59 3.1 PrimesoftheFormx 2+ky2 60 3.2 QuadraticIntegersandQuadraticForms 61 3.3 QuadraticFormsandEquivalence 63 vii Published online by Cambridge University Press viii Contents 3.4 CompositionofForms 66 3.5 FiniteAbelianGroups 68 3.6 TheChineseRemainderTheorem 71 3.7 AdditiveNotationforAbelianGroups 73 3.8 Discussion 74 4 RingsandFields 78 4.1 IntegersandFractions 79 4.2 DomainsandFieldsofFractions 82 4.3 PolynomialRings 83 4.4 AlgebraicNumberFields 86 4.5 FieldExtensions 89 4.6 TheIntegersofanAlgebraicNumberField 93 4.7 AnEquivalentDefinitionofAlgebraicInteger 96 4.8 Discussion 99 5 Ideals 104 5.1 “IdealNumbers” 105 5.2 Ideals 108 5.3 QuotientsandHomomorphisms 111 5.4 NoetherianRings 113 5.5 NoetherandtheAscendingChainCondition 116 5.6 CountableSets 119 5.7 Discussion 121 6 VectorSpaces 126 6.1 VectorSpaceBasisandDimension 127 6.2 Finite-DimensionalVectorSpaces 130 6.3 LinearMaps 134 6.4 AlgebraicNumbersasMatrices 136 6.5 TheTheoremofthePrimitiveElement 139 6.6 AlgebraicNumberFieldsandEmbeddingsinC 142 6.7 Discussion 144 7 DeterminantTheory 149 7.1 AxiomsfortheDeterminant 150 7.2 ExistenceoftheDeterminantFunction 153 7.3 DeterminantsandLinearEquations 156 7.4 BasisIndependence 159 7.5 TraceandNormofanAlgebraicNumber 161 7.6 Discriminant 164 7.7 Discussion 168 Published online by Cambridge University Press Contents ix 8 Modules 171 8.1 FromVectorSpacestoModules 172 8.2 AlgebraicNumberFieldsandTheirIntegers 174 8.3 IntegralBases 176 8.4 BasesandFreeModules 179 8.5 IntegersoveraRing 182 8.6 IntegralClosure 184 8.7 Discussion 186 9 IdealsandPrimeFactorization 189 9.1 ToDivideIstoContain 190 9.2 PrimeIdeals 192 9.3 ProductsofIdeals 194 9.4 PrimeIdealsinAlgebraicNumberRings 196 9.5 FractionalIdeals 197 9.6 PrimeIdealFactorization 199 9.7 InvertibilityandtheDedekindProperty 201 9.8 Discussion 204 References 211 Index 217 Published online by Cambridge University Press