Workbook in Higher Algebra - D. Surowski WW.pdf

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WorkbookinHigherAlgebra
DavidSurowski
DepartmentofMathematics
KansasStateUniversity
Manhattan,KS66506-2602,USA
dbski@math.ksu.edu
Contents
Acknowledgement ii
1GroupTheory 1
1.1ReviewofImportantBasics................... 1
1.2TheConceptofaGroupAction................. 5
1.3Sylow’sTheorem.........................12
1.4Examples:TheLinearGroups..................14
1.5AutomorphismGroups......................16
1.6TheSymmetricandAlternatingGroups............22
1.7TheCommutatorSubgroup...................28
1.8FreeGroups;GeneratorsandRelations ............36
2FieldandGaloisTheory 42
2.1Basics...............................42
2.2SplittingFieldsandAlgebraicClosure.............47
2.3GaloisExtensionsandGaloisGroups..............50
2.4SeparabilityandtheGaloisCriterion.............55
2.5BriefInterlude:theKrullTopology ..............61
2.6TheFundamentalTheoremofAlgebra ............62
2.7TheGaloisGroupofaPolynomial...............62
2.8TheCyclotomicPolynomials..................66
2.9SolvabilitybyRadicals......................69
2.10ThePrimitiveElementTheorem................70
3ElementaryFactorizationTheory 72
3.1Basics...............................72
3.2UniqueFactorizationDomains .................76
3.3NoetherianRingsandPrincipalIdealDomains........81
i
ii CONTENTS
3.4PrincipalIdealDomainsandEuclideanDomains.......84
4DedekindDomains 87
4.1AFewRemarksAboutModuleTheory.............87
4.2AlgebraicIntegerDomains....................91
4.3O E isaDedekindDomain....................96
4.4FactorizationTheoryinDedekindDomains..........97
4.5TheIdealClassGroupofaDedekindDomain.........100
4.6ACharacterizationofDedekindDomains...........101
5ModuleTheory 105
5.1TheBasicHomomorphismTheorems..............105
5.2DirectProductsandSumsofModules.............107
5.3ModulesoveraPrincipalIdealDomain............115
5.4CalculationofInvariantFactors.................119
5.5ApplicationtoaSingleLinearTransformation.........123
5.6ChainConditionsandSeriesofModules............129
5.7TheKrull-SchmidtTheorem...................132
5.8InjectiveandProjectiveModules................135
5.9SemisimpleModules.......................142
5.10Example:GroupAlgebras....................146
6RingStructureTheory 149
6.1TheJacobsonRadical......................149
7TensorProducts 154
7.1TensorProductasanAbelianGroup..............154
7.2TensorProductasaLeftS-Module...............158
7.3TensorProductasanAlgebra..................163
7.4Tensor,SymmetricandExteriorAlgebra............165
7.5TheAdjointnessRelationship..................172
AZorn’sLemmaandsomeApplications 175
Acknowledgement
ThepresentsetofnoteswasdevelopedasaresultofHigherAlgebracourses
thatItaughtduringtheacademicyears1987-88,1989-90and1991-92.The
distinctivefeatureofthesenotesisthatproofsarenotsupplied.There
aretworeasonsforthis.First,Iwouldhopethattheseriousstudentwho
reallyintendstomasterthematerialwillactuallytrytosupplymanyofthe
missingproofs.Indeed,Ihavetriedtobreakdowntheexpositioninsuch
awaythatbythetimeaproofiscalledfor,thereislittledoubtastothe
basicideaoftheproof.Therealreason,however,fornotsupplyingproofs
isthatifIhavetheproofsalreadyinhardcopy,thenmybasiclazinessoften
encouragesmenottospendanytimeinpreparingtopresenttheproofsin
class.Inotherwords,ifIcansimplyreadtheproofstothestudents,why
not?Ofcourse,themainreasonforthisisobvious;Ienduplookinglikea
fool.
Anyway,Iamthankfultothemanygraduatestudentswhocheckedand
critiquedthesenotes.IamparticularlyindebtedtoFrancisFungforhis
scoresofincisiveremarks,observationsandcorrections.Nontheless,these
notesareprobablyfarfromtheirfinalform;theywillsurelyundergomany
futurechanges,ifonlymotivitedbythesuggestionsofcolleaguesandfuture
graduatestudents.
Finally,IwishtosingleoutShanZhu,whohelpedwithsomeofthe
morelabor-intensiveaspectsofthepreparationofsomeoftheearlydrafts
ofthesenotes.Withouthishelp,theinertialdraginherentinmynature
wouldsurelyhavepreventedtheproductionofthissetofnotes.
DavidB.Surowski,
iii
Chapter1
GroupTheory
1.1ReviewofImportantBasics
Inthisshortsectionwegathertogethersomeofthebasicsofelementary
grouptheory,andatthesametimeestablishabitofthenotationwhichwill
beusedinthesenotes.Thefollowingtermsshouldbewell-understoodby
thereader(ifindoubt,consultanyelementarytreatmentofgrouptheory):
1 group,abeliangroup,subgroup,coset,normalsubgroup,quotientgroup,
orderofagroup,homomorphism,kernelofahomomorphism,isomorphism,
normalizerofasubgroup,centralizerofasubgroup,conjugacy,indexofa
subgroup,subgroupgeneratedbyasetofelementsDenotetheidentityele-
mentofthegroupGbye,andsetG # =G−{e}.IfGisagroupandifH
isasubgroupofG,weshallusuallysimplywriteHG.Homomorphisms
areusuallywrittenasleftoperators:thusif:G!G 0 isahomomorphism
ofgroups,andifg2G,writetheimageofginG 0 as(g).
Thefollowingisbasicinthetheoryoffinitegroups.
Theorem1.1.1(Lagrange’sTheorem)LetGbeafinitegroup,and
letHbeasubgroupofG.Then|H|divides|G|.
Thereadershouldbequitefamiliarwithboththestatement,aswellas
theproof,ofthefollowing.
Theorem1.1.2(TheFundamentalHomomorphismTheorem)Let
G, G 0 begroups,andassumethat:G!G 0 isasurjectivehomomorphism.
1 Many,ifnotmostofthesetermswillbedefinedbelow.
1

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