Algebra I: Textbook for Students of Mathematics

575 Pages·2016·5.64 MB·English

by  Alexey L. Gorodentsev

Alexey L. Gorodentsev
Algebra I
Textbook for Students of Mathematics
Algebra I
Alexey L. Gorodentsev
Algebra I
Textbook for Students of Mathematics
123
AlexeyL.Gorodentsev
FacultyofMathematics
NationalResearchUniversity
“HigherSchoolofEconomics”
Moscow,Russia
OriginallypublishedinRussianas“Algebra.Uchebnikdlyastudentov-matematikov.Chast’
1”,©MCCME2013
ISBN978-3-319-45284-5 ISBN978-3-319-45285-2 (eBook)
DOI10.1007/978-3-319-45285-2
LibraryofCongressControlNumber:2016959261
MathematicsSubjectClassification(2010):11.01,12.01,13.01,14.01,15.01,16.01,18.01,
20.01
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Preface
Thisisthefirstpartofanintensive2-yearcourseofalgebraforstudentsbeginning
aprofessionalstudyofhighermathematics.Thistextbookisbasedoncoursesgiven
attheIndependentUniversityofMoscowandattheFacultyofMathematicsinthe
NationalResearchUniversityHigherSchoolofEconomics.Inparticular,itcontains
alargenumberofexercisesthatwerediscussedinclass,someofwhichareprovided
with commentary and hints, as well as problems for independent solution, which
wereassignedashomework.Workingouttheexercisesisofcrucialimportancein
understandingthesubjectmatterofthisbook.
Moscow,Russia AlexeyL.Gorodentsev
v
Contents
1 Set-TheoreticandCombinatorialBackground.......................... 1
1.1 SetsandMaps........................................................ 1
1.1.1 Sets.......................................................... 1
1.1.2 Maps......................................................... 2
1.1.3 FibersofMaps.............................................. 3
1.2 EquivalenceClasses.................................................. 7
1.2.1 EquivalenceRelations...................................... 7
1.2.2 ImplicitlyDefinedEquivalences........................... 9
1.3 CompositionsofMaps............................................... 10
1.3.1 CompositionVersusMultiplication........................ 10
1.3.2 RightInverseMapandtheAxiomofChoice ............. 11
1.3.3 InvertibleMaps............................................. 12
1.3.4 TransformationGroups..................................... 12
1.4 Posets ................................................................. 13
1.4.1 PartialOrderRelations..................................... 13
1.4.2 Well-OrderedSets .......................................... 15
1.4.3 Zorn’sLemma .............................................. 15
ProblemsforIndependentSolutiontoChap.1............................. 16
2 IntegersandResidues...................................................... 19
2.1 Fields,Rings,andAbelianGroups.................................. 19
2.1.1 DefinitionofaField........................................ 19
2.1.2 CommutativeRings......................................... 21
2.1.3 AbelianGroups............................................. 21
2.1.4 SubtractionandDivision................................... 23
2.2 TheRingofIntegers ................................................. 24
2.2.1 Divisibility.................................................. 24
2.2.2 TheEquationaxCby D kandtheGreatest
CommonDivisorinZ...................................... 24
2.2.3 TheEuclideanAlgorithm .................................. 25
2.3 CoprimeElements.................................................... 26
vii
viii Contents
2.4 RingsofResidues.................................................... 27
2.4.1 ResidueClassesModulon ................................. 27
2.4.2 ZeroDivisorsandNilpotents .............................. 28
2.4.3 InvertibleElementsinResidueRings ..................... 28
2.4.4 ResidueFields .............................................. 29
2.5 DirectProductsofCommutativeGroupsandRings................ 30
2.6 Homomorphisms..................................................... 31
2.6.1 HomomorphismsofAbelianGroups...................... 31
2.6.2 KernelofaHomomorphism ............................... 32
2.6.3 GroupofHomomorphisms................................. 32
2.6.4 HomomorphismsofCommutativeRings.................. 33
2.6.5 HomomorphismsofFields................................. 34
2.7 ChineseRemainderTheorem........................................ 34
2.8 Characteristic......................................................... 35
2.8.1 PrimeSubfield.............................................. 35
2.8.2 FrobeniusEndomorphism.................................. 36
ProblemsforIndependentSolutiontoChap.2............................. 37
3 PolynomialsandSimpleFieldExtensions ............................... 41
3.1 FormalPowerSeries ................................................. 41
3.1.1 RingsofFormalPowerSeries ............................. 41
3.1.2 AlgebraicOperationsonPowerSeries .................... 42
3.1.3 Polynomials................................................. 43
3.1.4 DifferentialCalculus ....................................... 44
3.2 PolynomialRings .................................................... 46
3.2.1 Division..................................................... 46
3.2.2 CoprimePolynomials ...................................... 48
3.2.3 EuclideanAlgorithm ....................................... 48
3.3 RootsofPolynomials ................................................ 50
3.3.1 CommonRoots............................................. 50
3.3.2 MultipleRoots.............................................. 51
3.3.3 SeparablePolynomials..................................... 51
3.4 AdjunctionofRoots.................................................. 52
3.4.1 ResidueClassRings........................................ 52
3.4.2 AlgebraicElements......................................... 54
3.4.3 AlgebraicClosure .......................................... 55
3.5 TheFieldofComplexNumbers..................................... 55
3.5.1 TheComplexPlane......................................... 55
3.5.2 ComplexConjugation...................................... 58
3.5.3 Trigonometry ............................................... 58
3.5.4 RootsofUnityandCyclotomicPolynomials ............. 60
3.5.5 TheGaussianIntegers...................................... 62
3.6 FiniteFields .......................................................... 62
3.6.1 FiniteMultiplicativeSubgroupsinFields................. 62
3.6.2 DescriptionofAllFiniteFields............................ 63
Contents ix
3.6.3 QuadraticResidues......................................... 65
ProblemsforIndependentSolutiontoChap.3............................. 66
4 ElementaryFunctionsandPowerSeriesExpansions .................. 73
4.1 RingsofFractions.................................................... 73
4.1.1 Localization................................................. 73
4.1.2 FieldofFractionsofanIntegralDomain.................. 75
4.2 FieldofRationalFunctions.......................................... 76
4.2.1 SimplifiedFractions........................................ 76
4.2.2 PartialFractionExpansion................................. 77
4.2.3 PowerSeriesExpansionsofRationalFunctions.......... 79
4.2.4 LinearRecurrenceRelations............................... 80
4.3 LogarithmandExponential.......................................... 82
4.3.1 TheLogarithm.............................................. 83
4.3.2 TheExponential............................................ 83
4.3.3 PowerFunctionandBinomialFormula ................... 84
4.4 Todd’sSeriesandBernoulliNumbers............................... 88
4.4.1 ActionofQ(cid:2)d=dt(cid:3)onQŒt(cid:2) ................................. 88
4.4.2 BernoulliNumbers ......................................... 91
4.5 FractionalPowerSeries.............................................. 92
4.5.1 PuiseuxSeries .............................................. 92
4.5.2 Newton’sMethod........................................... 96
ProblemsforIndependentSolutiontoChap.4............................. 100
5 Ideals,QuotientRings,andFactorization ............................... 103
5.1 Ideals.................................................................. 103
5.1.1 DefinitionandExamples................................... 103
5.1.2 NoetherianRings ........................................... 104
5.2 QuotientRings ....................................................... 106
5.2.1 FactorizationHomomorphism............................. 106
5.2.2 MaximalIdealsandEvaluationMaps ..................... 107
5.2.3 PrimeIdealsandRingHomomorphismstoFields........ 108
5.2.4 FinitelyGeneratedCommutativeAlgebras................ 109
5.3 PrincipalIdealDomains ............................................. 109
5.3.1 EuclideanDomains......................................... 109
5.3.2 GreatestCommonDivisor ................................. 110
5.3.3 CoprimeElements.......................................... 111
5.3.4 IrreducibleElements........................................ 111
5.4 UniqueFactorizationDomains ...................................... 112
5.4.1 IrreducibleFactorization ................................... 112
5.4.2 PrimeElements............................................. 114
5.4.3 GCDinUniqueFactorizationDomains................... 115
5.4.4 PolynomialsoverUniqueFactorizationDomains ........ 116
x Contents
5.5 FactorizationofPolynomialswithRationalCoefficients........... 118
5.5.1 ReductionofCoefficients .................................. 118
5.5.2 Kronecker’sAlgorithm..................................... 119
ProblemsforIndependentSolutiontoChap.5............................. 120
6 Vectors....................................................................... 123
6.1 VectorSpacesandModules.......................................... 123
6.1.1 DefinitionsandExamples.................................. 123
6.1.2 LinearMaps................................................. 124
6.1.3 ProportionalVectors........................................ 125
6.2 BasesandDimension ................................................ 127
6.2.1 LinearCombinations....................................... 127
6.2.2 LinearDependence......................................... 130
6.2.3 BasisofaVectorSpace..................................... 132
6.2.4 Infinite-DimensionalVectorSpaces ....................... 134
6.3 SpaceofLinearMaps................................................ 135
6.3.1 KernelandImage........................................... 135
6.3.2 MatrixofaLinearMap..................................... 136
6.4 VectorSubspaces..................................................... 138
6.4.1 Codimension................................................ 138
6.4.2 LinearSpans................................................ 138
6.4.3 SumofSubspaces .......................................... 139
6.4.4 TranversalSubspaces....................................... 140
6.4.5 DirectSumsandDirectProducts .......................... 141
6.5 AffineSpaces......................................................... 142
6.5.1 DefinitionandExamples................................... 142
6.5.2 AffinizationandVectorization............................. 143
6.5.3 CenterofMass.............................................. 143
6.5.4 AffineSubspaces ........................................... 145
6.5.5 AffineMaps................................................. 148
6.5.6 AffineGroups............................................... 148
6.6 QuotientSpaces ...................................................... 149
6.6.1 QuotientbyaSubspace .................................... 149
6.6.2 QuotientGroupsofAbelianGroups....................... 150
ProblemsforIndependentSolutiontoChap.6............................. 151
7 Duality ....................................................................... 155
7.1 DualSpaces........................................................... 155
7.1.1 Covectors.................................................... 155
7.1.2 CanonicalInclusionV ,!V(cid:2)(cid:2) ............................ 158
7.1.3 DualBases.................................................. 158
7.1.4 Pairings...................................................... 160
7.2 Annihilators .......................................................... 161
Contents xi
7.3 DualLinearMaps.................................................... 164
7.3.1 PullbackofLinearForms .................................. 164
7.3.2 RankofaMatrix............................................ 165
ProblemsforIndependentSolutiontoChap.7............................. 167
8 Matrices...................................................................... 173
8.1 AssociativeAlgebrasoveraField................................... 173
8.1.1 DefinitionofAssociativeAlgebra ......................... 173
8.1.2 InvertibleElements......................................... 174
8.1.3 AlgebraicandTranscendentalElements................... 175
8.2 MatrixAlgebras...................................................... 175
8.2.1 MultiplicationofMatrices ................................. 175
8.2.2 InvertibleMatrices.......................................... 179
8.3 TransitionMatrices................................................... 180
8.4 GaussianElimination ................................................ 182
8.4.1 EliminationbyRowOperations ........................... 182
8.4.2 LocationofaSubspacewithRespecttoaBasis .......... 190
8.4.3 GaussianMethodforInvertingMatrices.................. 192
8.5 MatricesoverNoncommutativeRings.............................. 195
ProblemsforIndependentSolutiontoChap.8............................. 199
9 Determinants................................................................ 205
9.1 VolumeForms........................................................ 205
9.1.1 Volumeofann-DimensionalParallelepiped.............. 205
9.1.2 Skew-SymmetricMultilinearForms....................... 207
9.2 DigressiononParitiesofPermutations.............................. 208
9.3 Determinants ......................................................... 210
9.3.1 BasicPropertiesofDeterminants.......................... 211
9.3.2 DeterminantofaLinearEndomorphism.................. 214
9.4 GrassmannianPolynomials.......................................... 215
9.4.1 PolynomialsinSkew-CommutingVariables.............. 215
9.4.2 LinearChangeofGrassmannianVariables................ 216
9.5 LaplaceRelations .................................................... 217
9.6 AdjunctMatrix ....................................................... 220
9.6.1 RowandColumnCofactorExpansions ................... 220
9.6.2 MatrixInversion............................................ 221
9.6.3 Cayley–HamiltonIdentity.................................. 222
9.6.4 Cramer’sRules.............................................. 223
ProblemsforIndependentSolutiontoChap.9............................. 225
10 EuclideanSpaces ........................................................... 229
10.1 InnerProduct......................................................... 229
10.1.1 EuclideanStructure......................................... 229
10.1.2 LengthofaVector.......................................... 230
10.1.3 Orthogonality............................................... 230