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BasicAlgebra

DigitalSecondEditions

ByAnthonyW.Knapp

BasicAlgebra

AdvancedAlgebra

BasicRealAnalysis,

AdvancedRealAnalysis

AnthonyW.Knapp

BasicAlgebra

Alongwitha CompanionVolume AdvancedAlgebra

DigitalSecondEdition, 2016

Publishedbythe Author

EastSetauket, NewYork

nthonyW"Knapp ⌘UpperSheepPastureRoad

Title✓asiclgebra

Publishedbyirkh¬auseroston

Publishedbytheuthor

writtenpermissionfromtheauthor"

MediaInc"

iv

ToSusan

and

ToMyChildren,SarahandWilliam,

and

ToMyAlgebraTeachers:

RalphFox,JohnFraleigh,RobertGunning,

CONTENTS

Contentsofdvancedlgebrax

PrefacetotheSecondEditionxi

PrefacetotheFirstEditionxiii

ListofFiguresxvii

DependenceAmongChaptersxix

StandardNotationxx

GuidefortheReaderxxi

I.PRELIMINARIESABOUTTHEINTEGERS,

POLYNOMIALS,ANDMATRICES1

1.DivisionandEuclideanAlgorithms1

2.UniqueFactorizationofIntegers4

3.UniqueFactorizationofPolynomials9

4.PermutationsandTheirSigns15

5.RowReduction19

6.MatrixOperations24

7.Problems30

II.VECTORSPACESOVERQ,R,AND⌧33

1.Spanning,LinearIndependence,andBases33

2.VectorSpacesDefinedbyMatrices38

3.LinearMaps42

4.DualSpaces50

5.QuotientsofVectorSpaces54

7.Determinants65

9.BasesintheInfinite-DimensionalCase78

10.Problems82

III.INNER-PRODUCTSPACES89

1.InnerProductsandOrthonormalSets89

2.Adjoints99

3.SpectralTheorem105

4.Problems112

vii viiiContents

IV.GROUPSANDGROUPACTIONS117

1.GroupsandSubgroups118

2.QuotientSpacesandHomomorphisms129

3.DirectProductsandDirectSums135

4.RingsandFields141

5.PolynomialsandVectorSpaces148

6.GroupActionsandExamples159

7.SemidirectProducts167

8.SimpleGroupsandCompositionSeries171

10.SylowTheorems185

11.CategoriesandFunctors189

12.Problems200

V.THEORYOFASINGLELINEARTRANSFORMATION211

1.Introduction211

3.CharacteristicandMinimalPolynomials218

4.ProjectionOperators226

5.PrimaryDecomposition228

6.JordanCanonicalForm231

7.ComputationswithJordanForm238

8.Problems241

VI.MULTILINEARALGEBRA248

1.BilinearFormsandMatrices249

2.SymmetricBilinearForms253

3.AlternatingBilinearForms256

4.HermitianForms258

5.GroupsLeavingaBilinearFormInvariant260

6.TensorProductofTwoVectorSpaces263

7.TensorAlgebra277

8.SymmetricAlgebra283

9.ExteriorAlgebra291

10.Problems295

VII.ADVANCEDGROUPTHEORY306

1.FreeGroups306

2.SubgroupsofFreeGroups317

3.FreeProducts322

4.GroupRepresentations329

Contentsix

VII.ADVANCEDGROUPTHEORY(Continued)

5.Burnside'sTheorem345

6.ExtensionsofGroups347

7.Problems360

VIII.COMMUTATIVERINGSANDTHEIRMODULES370

1.ExamplesofRingsandModules370

2.IntegralDomainsandFieldsofFractions381

3.PrimeandMaximalIdeals384

4.UniqueFactorization387

5.Gauss'sLemma393

6.FinitelyGeneratedModules399

7.OrientationforAlgebraicNumberTheoryand

AlgebraicGeometry411

9.IntegralClosure420

10.LocalizationandLocalRings428

11.DedekindDomains437

12.Problems443

IX.FIELDSANDGALOISTHEORY452

1.AlgebraicElements453

2.ConstructionofFieldExtensions457

3.FiniteFields461

4.AlgebraicClosure464

6.SeparableExtensions474

7.NormalExtensions481

8.FundamentalTheoremofGaloisTheory484

NonsolvableGaloisGroup493

12.ConstructionofRegularPolygons499

GaloisGroup506

14.ProofThatIsTranscendental515

15.NormandTrace519

16.SplittingofPrimeIdealsinExtensions526

17.TwoToolsforComputingGaloisGroups532

18.Problems539

xContents

X.MODULESOVERNONCOMMUTATIVERINGS553

1.SimpleandSemisimpleModules553

2.CompositionSeries560

3.ChainConditions565

4.HomandEndforModules567

5.TensorProductforModules574

6.ExactSequences583

7.Problems587

APPENDIX593

A1.SetsandFunctions593

A2.EquivalenceRelations599

A3.RealNumbers601

A4.ComplexNumbers604

A5.PartialOrderingsandZorn'sLemma605

A6.Cardinality610

HintsforSolutionsofProblems615

SelectedReferences715

IndexofNotation717

Index721

CONTENTSOFDVN⌧EDLGER

I.TransitiontoModernNumberTheory

II.Wedderburn-ArtinRingTheory

III.BrauerGroup

IV.HomologicalAlgebra

V.ThreeTheoremsinAlgebraicNumberTheory

VI.ReinterpretationwithAdelesandIdeles

VII.InfiniteFieldExtensions

VIII.BackgroundforAlgebraicGeometry

IX.TheNumberTheoryofAlgebraicCurves

X.MethodsofAlgebraicGeometry

PREFACETOTHESECONDEDITION

applicationsofthetheory. elsewhere. earlierproofhavinghadagap. minantsandareasorvolumes, linearmappings, xi xiiPrefacetotheSecondEdition fields. asareminderofhow hadbeenleftas

Xratherthan

AdvancedAlgebra.

donebytheprogramTexturesusing AMS-T E

X,andthefiguresweredrawnwith

Mathematica.

"W"KNPPJanuary2016

PREF⌧ETOTHEFIRSTEDITION

multilinearalgebra. numbertheoryandgeometry. xiii xivPrefacetotheFirstEdition numberstoprovethat groups. shedefinedtheingredientsofwhatwasthencalled"modernalgebra" - the

PrefacetotheFirstEditionxv

andconstructionswithmodules.

ChapterIX.

xviPrefacetotheFirstEdition reallynecessary. word"

PROOF"or"

symboltomarktheendofthatproof.

Thetypesettingwasby

AMS-T E

X,andthefiguresweredrawnwithMathematica.

ofknowncorrectionsonmyownWebpage. "W"KNPPAugust2006

LISTOFFIGURES

⇣"⌘"Thevectorspaceoflines v⌥ UinR paralleltoagivenline

Uthroughtheorigin

suchthateachpair orthogonalcomponent groupbyanormalsubgroup⌘ groups⌘ byanideal⌘ nindeterminates⌘ ""squarediagram⌘ contravariantfunctor⌘ xvii xviiiListofFigures

TX}⇣◆

""Equivalentgroupextensions⇣ G

Rmodule

""Factorizationof

Rhomomorphismsviaaquotientof

Rmodules

RG⌘

R "◆"Realpointsofthecurve y⇣ ⇧x⌘ ⌫x⇧x⌥⌘ Rat

S⌘

xcoordinatesunder

Rmodule

andaleft

Rmodule

DEPENDENCEAMONGCHAPTERS

thatisnotindicatedbythechart. I,II III

IV.1-IV.6

IV.7-IV.11V

VIIVIVIII.1-VIII.6

XIX.1-IX.13VIII.7-VIII.11

IX.14-IX.17

xix

STNDRDNOTTION

ItemMeaning

Sor|

S|numberofelementsin

Semptyset

x E|

P}thesetof

xin

Esuchthat

Pholds

Eccomplementoftheset

F⇠E⌅

F⇠E

Eunion⇠intersectionofthesets

EE

F⇠E

FEiscontainedin

F⇠

Econtains

F⇠E⌅

FEproperlycontainedin

F⇠properlycontains

FE⇥

F⇠

s

SXsproductsofsets

⇧a⌘ ⇠"""⇠an⌫⇠{ a⌘ ⇠"""⇠an}ordered n!tuple⇠unordered n!tuple f✓

E⇧

F⇠x⌫⇧

f⌃ gor fg⇠f⌅

Ecompositionof

gfollowedby f⇠restrictionto

Ef⇧á

⇠y⌫thefunction x⌫⇧ ⇧E⌫directandinverseimageofaset i i⌥ j nk binomialcoefÞcient npositive⇠ nnegative or x]greatestinteger⇠ xif xisreal Re z⇠Im zrealandimaginarypartsofcomplex zø zcomplexconjugateof z| z|absolutevalueof z⌘multiplicativeidentity ⌘or

Iidentitymatrixoroperator

Xidentityfunctionon

XQ n⇠R n⇠⌧ nspacesofcolumnvectors diag ⇧a⌘ ⇠"""⇠an⌫diagonalmatrix isisomorphicto⇠isequivalentto xx

GUIDEFORTHEREADER

headed. fewsectionsofChapterVIII. ofchapters. xxi xxiiGuidefortheReader inChapterV.

GuidefortheReaderxxiii

sametimeastheobjects. materialoncategorytheory. xxivGuidefortheReader groups. theme - similaritiesbetweentheintegersandcertainpolynomialrings.Section7 aboutwhichregular constructibleregular istranscendental

VIandVIII,butnotfromChapterIX.

BasicAlgebra

CHAPTERI

andMatrices permutations,andmatrixalgebra. function.

ChapterIX.

thesignofaproductistheproductofthesigns.

1.DivisionandEuclideanAlgorithms

orderinginZ.

Afactorofaninteger

nisanonzerointeger ksuchthat n klforsome integer l.Inthiscasewesayalsothat kdivides n,that kisadivisorof n,and that nisamultipleof k.Wewrite k| nforthisrelationship.If nisnonzero,any productformula n kl1 lrisafactorizationof n.AunitinZisadivisor 1 n klof nontrivialifneither knor lisaunit.Aninteger p⇢1issaidtobeprimeifit hasnonontrivialfactorization p kl. 1 following.

Proposition1.1(divisionalgorithm).If

aand bareintegerswith thereexistuniqueintegers qand rsuchthat a bq⌥ rand0⌅ r⇡| b|.

PROOF.Possiblyreplacing

qby q,wemayassumethat b⇢0.Theintegers nwith bn⌅ aareboundedaboveby| a|,andthereexistssuchan n,namely n | n q.Set r a bq.Then0⌅ rand a bq⌥ r.If r b,then r b0saysthat a b⇧q⌥1 ⇧r b⌫ b⇧q⌥1 ⌫.Theinequality q⌥1 ⇢qcontradictsthe maximalityof q,andweconcludethat r⇡b.Thisprovesexistence.

Foruniquenesswhen

b⇢0,suppose a bq1 r1 bq2 r2 .Subtracting, weobtain b⇧q1 q2 r2 r1 with| r2 r1 ⇡b,andthisisacontradiction unless r2 r1 0. Let aand aand bisthelargestinteger d⇢0suchthat d| aand d| b.Letusseeexistence.

Theinteger1divides

aand b.If b,forexample,isnonzero,thenanysuch dhas| d|⌅| d GCD ⇧a⇠b⌫.

Letussupposethat

aand buntiltheremainder term rdisappears: a bq1 r1 ⇠0⌅ r1 ⇡b⇠b r1 q2 r2 ⇠0⌅ r2 ⇡r1 ⇠r1 r2 q3 r3 ⇠0⌅ r3 ⇡r2 ⇠"""rn2 rn1 qn⌥quotesdbs_dbs48.pdfusesText_48

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