Algebra: Notes From The Underground

490 Pages·2021·3.859 MB·English

by  Paolo Aluffi

Algebra
From rings to modules to groups to fields, this undergraduate introduction to abstract
algebrafollowsanunconventional path.Thetextemphasizesamodernperspectiveon
the subject, with gentle mentions of the unifying categorical principles underlying the
variousconstructionsandtheroleofuniversalproperties.Akeyfeatureisthetreatment
of modules, including a proof of the classification theorem for finitely generated
modules over Euclidean domains. Noetherian modules and some of the language of
exact complexes are introduced. In addition, standard topics—such as the Chinese
RemainderTheorem,theGaussLemma,theSylowTheorems,simplicityofalternating
groups, standard results on field extensions, and the Fundamental Theorem of Galois
Theory—are all treated in detail. Students will appreciate the text’s conversational
style,400+exercises,appendixwithcompletesolutionstoaround150ofthemaintext
problems,andappendixwithgeneralbackgroundonbasiclogicandnaïvesettheory.
Paolo Aluffi is Professor of Mathematics at Florida State University. Aluffi earned
a Ph.D. from Brown University with a dissertation on the enumerative geometry of
plane cubic curves, under the supervision of William Fulton. His research interests
are in algebraic geometry, particularly intersection theory and its application to the
theoryofsingularitiesandconnectionswiththeoreticalphysics.Hehasauthoredabout
70 research publications and given lectures on his work in 15 countries. Beside Notes
fromtheUnderground,hehaspublishedagraduate-leveltextbookinalgebra(Algebra:
Chapter 0, AMS) and a mathematics book for the general public, in Italian (Fare
matematica,AracneEditrice).
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Algebra
PAOLO ALUFFI
FloridaStateUniversity
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108958233
DOI:10.1017/9781108955911
©PaoloAluffi2021
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2021
PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall
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ISBN978-1-108-95823-3Paperback
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Contents
Introduction pageix
BeforeweBegin xiii
PartI Rings 1
1 TheIntegers 3
1.1 TheWell-OrderingPrincipleandInduction 3
1.2 ‘DivisionwithRemainder’inZ 6
1.3 GreatestCommonDivisors 7
1.4 TheFundamentalTheoremofArithmetic 12
Exercises 20
2 ModularArithmetic 22
2.1 EquivalenceRelationsandQuotients 22
2.2 Congruencemodn 23
2.3 AlgebrainZ/nZ 26
2.4 PropertiesoftheOperations+,·onZ/nZ 29
2.5 Fermat’sLittleTheorem,andtheRSAEncryptionSystem 34
Exercises 39
3 Rings 41
3.1 DefinitionandExamples 41
3.2 BasicProperties 47
3.3 SpecialTypesofRings 51
Exercises 57
4 TheCategoryofRings 59
4.1 CartesianProducts 59
4.2 Subrings 61
4.3 RingHomomorphisms 64
4.4 IsomorphismsofRings 69
Exercises 75
vi Contents
5 CanonicalDecomposition,Quotients,andIsomorphismTheorems 77
5.1 Rings:CanonicalDecomposition,I 77
5.2 KernelsandIdeals 79
5.3 QuotientRings 83
5.4 Rings:CanonicalDecomposition,II 89
5.5 TheFirstIsomorphismTheorem 91
5.6 TheChineseRemainderTheorem 93
5.7 TheThirdIsomorphismTheorem 100
Exercises 103
6 IntegralDomains 106
6.1 PrimeandMaximalIdeals 106
6.2 PrimesandIrreducibles 110
6.3 EuclideanDomainsandPIDs 112
6.4 PIDsandUFDs 117
6.5 TheFieldofFractionsofanIntegralDomain 120
Exercises 126
7 PolynomialRingsandFactorization 129
7.1 Fermat’sLastTheoremforPolynomials 129
7.2 ThePolynomialRingwithCoefficientsinaField 131
7.3 IrreducibilityinPolynomialRings 136
7.4 IrreducibilityinQ[x]andZ[x] 139
7.5 IrreducibilityTestsinZ[x] 143
Exercises 148
PartII Modules 151
8 ModulesandAbelianGroups 153
8.1 VectorSpacesandIdeals,Revisited 153
8.2 TheCategoryofR-Modules 158
8.3 Submodules,DirectSums 161
8.4 CanonicalDecompositionandQuotients 164
8.5 IsomorphismTheorems 168
Exercises 171
9 ModulesoverIntegralDomains 174
9.1 FreeModules 174
9.2 ModulesfromMatrices 181
9.3 FinitelyGeneratedvs.FinitelyPresented 187
9.4 VectorSpacesareFreeModules 190
9.5 FinitelyGeneratedModulesoverEuclideanDomains 193
9.6 LinearTransformationsandModulesoverk[t] 197
Exercises 199
Contents vii
10 AbelianGroups 202
10.1 TheCategoryofAbelianGroups 202
10.2 CyclicGroups,andOrdersofElements 207
10.3 TheClassificationTheorem 212
10.4 Fermat’sTheoremonSumsofSquares 217
Exercises 223
PartIII Groups 227
11 Groups—Preliminaries 229
11.1 GroupsandtheirCategory 229
11.2 WhyGroups?ActionsofaGroup 235
11.3 Cyclic,Dihedral,Symmetric,FreeGroups 241
11.4 CanonicalDecomposition,Normality,andQuotients 253
11.5 IsomorphismTheorems 260
Exercises 264
12 BasicResultsonFiniteGroups 267
12.1 TheIndexofaSubgroup,andLagrange’sTheorem 267
12.2 StabilizersandtheClassEquation 269
12.3 ClassificationandSimplicity 274
12.4 Sylow’sTheorems:Statements,Applications 276
12.5 Sylow’sTheorems:Proofs 280
12.6 SimplicityofA 282
n
12.7 SolvableGroups 286
Exercises 289
PartIV Fields 293
13 FieldExtensions 295
13.1 FieldsandHomomorphismsofFields 295
13.2 FiniteExtensionsandtheDegreeofanExtension 299
13.3 SimpleExtensions 301
13.4 AlgebraicExtensions 307
13.5 Application:‘GeometricImpossibilities’ 309
Exercises 314
14 NormalandSeparableExtensions,andSplittingFields 317
14.1 SimpleExtensions,Again 317
14.2 SplittingFields 320
14.3 NormalExtensions 326
14.4 SeparableExtensions;andSimpleExtensionsOnceAgain 328
14.5 Application:FiniteFields 333
Exercises 337
viii Contents
15 GaloisTheory 340
15.1 GaloisGroupsandGaloisExtensions 340
15.2 CharacterizationofGaloisExtensions 345
15.3 TheFundamentalTheoremofGaloisTheory 350
15.4 GaloisGroupsofPolynomials 357
15.5 SolvingPolynomialEquationsbyRadicals 361
15.6 OtherApplications 370
Exercises 373
AppendixA Background 377
AppendixB SolutionstoSelectedExercises 402
IndexofDefinitions 461
IndexofTheorems 463
SubjectIndex 465