Algebraic Number Theory for Beginners: Following a Path From Euclid to Noether

243 Pages·2022·3.287 MB·English

by  John Stillwell

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
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Informationonthistitle:www.cambridge.org/9781316518953
DOI:10.1017/9781009004138
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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
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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
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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
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