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首页 抽象代数初论-Rotman

抽象代数初论-Rotman.pdf

抽象代数初论-Rotman

凌云
2011-04-15 0人阅读 举报 0 0 暂无简介

简介:本文档为《抽象代数初论-Rotmanpdf》,可适用于人文社科领域

AFIRSTCOURSEINABSTRACTALGEBRAThirdEditionJOSEPHJROTMANUniversityofIllinoisatUrbanaChampaignPRENTICEHALL,UpperSaddleRiver,NewJerseyTomytwowonderfulkids,DannyandElla,whomIloveverymuchContentsSpecialNotationiContentsvPrefacetotheThirdEditionviiiChapterNumberTheorySectionInductionSectionBinomialCoefficientsSectionGreatestCommonDivisorsSectionTheFundamentalTheoremofArithmeticSectionCongruencesSectionDatesandDaysChapterGroupsISectionSomeSetTheoryFunctionsEquivalenceRelationsSectionPermutationsSectionGroupsSymmetrySectionSubgroupsandLagrange’sTheoremSectionHomomorphismsSectionQuotientGroupsSectionGroupActionsSectionCountingwithGroupsChapterCommutativeRingsIvviCONTENTSSectionFirstPropertiesSectionFieldsSectionPolynomialsSectionHomomorphismsSectionGreatestCommonDivisorsEuclideanRingsSectionUniqueFactorizationSectionIrreducibilitySectionQuotientRingsandFiniteFieldsSectionOfficers,Magic,Fertilizer,andHorizonsOfficersMagicFertilizerHorizonsChapterLinearAlgebraSectionVectorSpacesGaussianEliminationSectionEuclideanConstructionsSectionLinearTransformationsSectionDeterminantsSectionCodesBlockCodesLinearCodesChapterFieldsSectionClassicalFormulasVie`te’sCubicFormulaSectionInsolvabilityoftheGeneralQuinticFormulasandSolvabilitybyRadicalsTranslationintoGroupTheorySectionEpilogChapterGroupsIISectionFiniteAbelianGroupsSectionTheSylowTheoremsSectionOrnamentalSymmetryChapterCommutativeRingsIISectionPrimeIdealsandMaximalIdealsSectionUniqueFactorizationSectionNoetherianRingsCONTENTSviiSectionVarietiesSectionGro¨bnerBasesMonomialOrdersGeneralizedDivisionAlgorithmGro¨bnerBasesAppendixAInequalitiesAppendixBPseudocodesHintsforSelectedExercisesBibliographyIndexPrefacetotheThirdEditionAFirstCourseinAbstractAlgebraintroducesgroupsandcommutativeringsGrouptheorywasinventedbyEGaloisintheearlys,whenheusedgroupstocompletelydeterminewhentherootsofpolynomialscanbefoundbyformulasgeneralizingthequadraticformulaNowadays,groupsaretheprecisewaytodiscussvarioustypesofsymmetry,bothingeometryandelsewhereBesidesintroducingGalois’ideas,wealsoapplygroupstosomeintricatecountingproblemsaswellastotheclassificationoffriezesintheplaneCommutativeringsprovidethepropercontextinwhichtostudynumbertheoryaswellasmanyaspectsofthetheoryofpolynomialsForexample,generalizationsofideassuchasgreatestcommondivisorandmodulararithmeticextendeffortlesslytopolynomialringsoverfieldsApplicationsincludepublicaccesscodes,finitefields,magicsquares,Latinsquares,andcalendarsWethenconsidervectorspaceswithscalarsinarbitraryfields(notjustthereals),andthisstudyallowsustosolvetheclassicalGreekproblemsconcerningangletrisection,doublingthecube,squaringthecircle,andconstructionofregularngonsLinearalgebraoverfinitefieldsisappliedtocodes,showinghowonecanaccuratelydecodemessagessentoveranoisychannel(forexample,photographssenttoEarthfromMarsorfromSaturn)Here,oneseesfinitefieldsbeingusedinanessentialwayInChapter,wegivetheclassicalformulasfortherootsofcubicandquarticpolynomials,afterwhichbothgroupsandcommutativeringstogetherareusedtoproveGalois’theorem(polynomialswhoserootsareobtainablebysuchformulashavesolvableGaloisgroups)andAbel’stheorem(thereisnogeneralizationoftheseformulastopolynomialsofhigherdegree)ThisisonlyanintroductiontoGaloistheoryreaderswishingtolearnmoreofthisbeautifulsubjectwillhavetoseeamoreadvancedtextForthosereaderswhoseappetiteshavebeenwhettedbytheseresults,thelasttwochaptersinvestigategroupsandringsfurther:weprovethebasistheoremforfiniteabeliangroupsandtheSylowtheorems,andweintroducethestudyofpolynomialsinseveralvariables:varietiesHilbert’sbasisviiiPREFACETOTHETHIRDEDITIONixtheorem,thestellensatz,andalgorithmicmethodsassociatedwithGro¨bnerbasesLetmementionsomenewfeaturesofthiseditionIhaverewrittenthetext,addingmoreexercises,andtryingtomaketheexpositionmoresmoothThefollowingchangesinformatshouldmakethebookmoreconvenienttouseEveryexerciseexplicitlycitedelsewhereinthetextismarkedbyanasteriskmoreover,everycitationgivesthepagenumberonwhichthecitedexerciseappearsHintsforcertainexercisesareinasectionattheendofthebooksothatreadersmayconsiderproblemsontheirownbeforereadinghintsOnenumberingsystemenumeratesalllemmas,theorems,propositions,corollaries,andexamples,sothatfindingbackreferencesiseasyThereareseveralpagesofSpecialNotation,givingpagenumberswherenotationisintroducedToday,abstractalgebraisviewedasachallengingcoursemanybrightstudentsseemtohaveinordinatedifficultylearningitCertainly,theymustlearntothinkinanewwayAxiomaticreasoningmaybenewtosomeothersmaybemorevisuallyorientedSomestudentshaveneverwrittenproofsothersmayhaveoncedoneso,buttheirskillshaveatrophiedfromlackofuseButnoneoftheseobstaclesadequatelyexplainstheobserveddifficultiesAfterall,thesameobstaclesexistinbeginningrealanalysiscourses,butmoststudentsinthesecoursesdolearnthematerial,perhapsaftersomeearlystrugglingHowever,thedifficultyofstandardalgebracoursespersists,whethergroupsaretaughtfirst,whetherringsaretaughtfirst,orwhethertextsarechangedIbelievethatamajorcontributingfactortothedifficultyinlearningabstractalgebraisthatbothgroupsandringsareintroducedinthefirstcourseassoonasastudentbeginstobecomfortablewithonetopic,itisdroppedtostudytheotherFurthermore,ifoneleavesgrouptheoryorcommutativeringtheorybeforesignificantapplicationscanbegiven,thenstudentsareleftwiththefalseimpressionthatthetheoryiseitherofnorealvalueor,morelikely,thatitcannotbeappreciateduntilsomefutureindefinitetime(Imagineabeginninganalysiscourseinwhichbothrealandcomplexanalysisareintroducedinonesemester)Ifalgebraistaughtasaoneyear(twosemester)course,thereisnolongeranyreasontocrowdbothtopicsintothefirstcourse,andatruer,moreattractive,pictureofalgebraispresentedThisoptionismorepracticaltodaythaninthepast,forthemanyapplicationsofabstractalgebrahaveincreasedthenumbersofinterestedstudents,manyofwhomareworkinginotherdisciplinesIhaverewrittenthistextfortwoaudiencesThisneweditioncanserveasatextforthosewhowishtocontinueteachingthecurrentlypopulararrangementofintroducingbothgroupsandringsinthefirstsemesterAsusual,onebeginsbycoveringmostofChapter,afterwhichonechoosesselectedpartsofChaptersand,dependingonwhethergroupsorcommutativeringsaretaughtfirstChaptersandhavebeenrewritten,andtheyarenowessentiallyindependentxPREFACETOTHETHIRDEDITIONofoneanother,sothatthisbookmaybeusedforeitherorderofpresentation(Asanaside,Idisagreewiththecurrentreceivedwisdomthatdoinggroupsfirstismoreefficientthandoingringsfirstforexample,thepresentversionofChapterisaboutthesamelengthasitsearlierversions)ThereisamplematerialinthebooksothatitcanfurtherserveasatextforasequelcourseaswellLetmenowaddressasecondaudience:thosewillingtotryanewapproachMyownideasaboutteachingabstractalgebrahavechanged,andInowthinkthatatwosemestercourseinwhichonlyoneofgroupsorringsistaughtinthefirstsemester,isbestIrecommendaoneyearcoursewhosefirstsemestercoversnumbertheoryandcommutativerings,andwhosesecondsemestercoverslinearalgebraandgrouptheoryInmoredetail,thefirstsemestershouldtreattheusualselectionofarithmetictheoremsinChapter:divisionalgorithmgcd’seuclideanalgorithmuniquefactorizationcongruenceChineseremaindertheoremContinuewithSection:functionsinversefunctionsequivalencerelations,andthencommutativeringsinChapter:fractionfieldsofdomainsgeneralizationsofarithmetictheoremstopolynomialsidealsintegersmodmisomorphismtheoremssplittingfields,existenceoffinitefields,magicsquares,orthogonalLatinsquaresOnecouldinsteadcontinueoninChapter,coveringgrouptheoryinsteadofcommutativerings,butIthinkthatdoingcommutativeringsfirstismoreuserfriendlyItisnaturaltopassfrom�tokx,andonecanwatchhowthenotionofidealdevelopsfromatechniqueshowingthatgcd’sarelinearcombinationsintoanimportantideaForthesecondsemester,IrecommendbeginningwithportionsofChapter:linearalgebraoverarbitraryfields:invarianceofdimensionrulercompassconstructionsmatricesandlineartransformationsdeterminantsovercommutativeringsMostofthismaterialcanbedonequicklyifthestudentshavecompletedanearlierlinearalgebracoursetreatingvectorspacesover�Iftimepermits,onecanreadthesectiononcodes,whichculminateswithaproofthatReedSolomoncodescanbedecodedTheremainderofthesemestershoulddiscussgroups,asinChapter:permutationssymmetriesofplanarfiguresLagrange’stheoremisomorphismtheoremsgroupactionsBurnsidecountingandfriezegroups,asinChapterIfthereisnotampletimetocovercodesandfriezegroups,thesesectionsareappropriatespecialprojectsforinterestedstudentsIpreferthisorganizationandpresentation,andIbelievethatitisanimprovementoverthatofstandardcoursesGivingtheetymologyofmathematicaltermsisrarelydoneLetmeexplain,withananalogy,whyIhaveincludedderivationsofmanytermsTherearemanyvariationsofstandardpokergamesand,inmypokergroup,thedealerannouncesthegameofhischoicebynamingitNowsomenamesarebetterthanothersForexample,“LittleRed”isagameinwhichone’ssmallestredcardiswildthisisagoodnamebecauseitremindstheplayersofitsdistinctivefeatureOnthePREFACETOTHETHIRDEDITIONxiotherhand,“Aggravation”isnotsuchagoodname,forthoughitis,indeed,suggestive,thenamedoesnotdistinguishthisparticulargamefromseveralothersMosttermsinmathematicshavebeenwellchosentherearemorerednamesthanaggravatingonesAnexampleofagoodnameisevenpermutation,forapermutationisevenifitisaproductofanevennumberoftranspositionsAnotherexampleofagoodtermistheparallelogramlawdescribingvectoradditionButmanygoodnames,clearwhentheywerechosen,arenowobscurebecausetheirrootsareeitherinanotherlanguageorinanotherdisciplineThetrigonometrictermstangentandsecantaregoodnamesforthoseknowingsomeLatin,buttheyareobscureotherwise(seeadiscussionoftheiretymologyonpage)ThetermmathematicsisobscureonlybecausemostofusdonotknowthatitcomesfromtheclassicalGreekwordmeaning“tolearn”ThetermcorollaryisdoublyobscureitcomesfromtheLatinwordmeaning“flower,”butwhyshouldsometheoremsbecalledflowersAplausibleexplanationisthatitwascommon,inancientRome,togiveflowersasgifts,andsoacorollaryisagiftbequeathedbyatheoremThetermtheoremcomesfromtheGreekwordmeaning“towatch”or“tocontemplate”(theatrehasthesameroot)itwasusedbyEuclidwithitspresentmeaningThetermlemmacomesfromtheGreekwordmeaning“taken”or“received”itisastatementthatistakenforgranted(forithasalreadybeenproved)inthecourseofprovingatheoremIbelievethatetymologyoftermsisworthwhile(andinteresting!),foritoftenaidsunderstandingbyremovingunnecessaryobscurityInadditiontothankingagainthosewhohelpedmewiththefirsttwoeditions,itisapleasuretothankGeorgeBergmanandChrisHeilfortheirvaluablecommentsonthesecondeditionIalsothankIwanDuursma,RobertFriedman,BlairFGoodlin,DieterKoller,FatmaIremKoprulu,JPeterMay,LeonMcCulloh,ArnoldMiller,BrentBSolie,andJohnWetzelJosephJRotmanxiiPREFACETOTHETHIRDEDITIONNumberTheoryINDUCTIONTherearemanystylesofproof,andmathematicalinductionisoneofthemWebeginbysayingwhatmathematicalinductionisnotInthenaturalsciences,inductivereasoningistheassertionthatafreqentlyobservedphenomenonwillalwaysoccurThus,onesaysthattheSunwillrisetomorrowmorningbecause,fromthedawnoftime,theSunhasriseneverymorningThisisnotalegitimatekindofproofinmathematics,foreventhoughaphenomenonhasbeenobservedmanytimes,itneednotoccurforeverHowever,inductivereasoningisstillvaluableinmathematics,asitisinnaturalscience,becauseseeingpatternsindataoftenhelpsinguessingwhatmaybetrueingeneralOntheotherhand,areasonableguessmaynotbecorrectForexample,whatisthemaximumnumberofregionsintowhich�(dimensionalspace)canbedividedbynplanesTwononparallelplanescandivide�intoregions,andthreeplanescandivide�intoregions(octants)Forsmallern,wenotethatasingleplanedivides�intoregions,whileifn=,then�isnotdividedatall:thereisregionForn=,,,,themaximumnumberofregionsisthus,,,,anditisnaturaltoguessthatnplanescanbechosentodivide�intonregionsButitturnsoutthatanyfourchosenplanescandivide�intoatmostregions!Beforeproceedingfurther,letusmakesurethatweagreeonthemeaningofsomestandardtermsAnintegerisoneofthenumbers,,−,,−,,thesetofalltheintegersisdenotedby�(fromtheGermanZahlmeaningnumber):�={,,−,,−,,}Thenaturalnumbersconsistsofallthoseintegersnforwhichn≥:�={nin�:n≥}={,,,,}NUMBERTHEORYCHDefinitionAnintegerdisadivisorofanintegernifn=daforsomeintegeraAnintegerniscalledprimeifn≥anditsonlydivisorsare±and±nanintegerniscalledcompositeifitisnotprimeIfapositiveintegerniscomposite,thenithasafactorizationn=ab,wherea<nandb<narepositiveintegerstheinequalitiesarepresenttoeliminatetheuninterestingfactorizationn=n×Thefirstfewprimesare,,,,,,,,,,,,,thatthissequenceneverendsisprovedinCorollaryConsidertheassertionthatf(n)=n−nisprimeforeverypositiveintegernEvaluatingf(n)forn=,,,,givesthenumbers,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Itistedious,butnotverydifficult,toshowthateveryoneofthesenumbersisprime(seeProposition)Inductivereasoningpredictsthatallthenumbersoftheformf(n)areprimeButthenextnumber,f()=,isnotprime,forf()=−=,whichisobviouslycompositeThus,inductivereasoningisnotappropriateformathematicalproofsHereisanevenmorespectacularexample(whichIfirstsawinanarticlebyWSierpinski)Recallthatperfectsquaresarenumbersoftheformn,wherenisanintegerthefirstfewperfectsquaresare,,,,,,,Foreachn≥,considerthestatementS(n):nisnotaperfectsquareThenthstatement,S(n),istrueformanyninfact,thesmallestnumbernforwhichS(n)isfalseisn=,,,,,,,,,≈

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