Advanced Algebra - Mathematics and Statistics

1 ,withanappendix ElementaryComplexAnalysis AdvancedRealAnalysisAnthony W. KnappAdvanced AlgebraAlong with a Companion VolumeBasic AlgebraDigital Second Edition, 2016 Published by the AuthorEast Setauket, New , aknappTitle:BasicAlgebraCover:Constructi onofaregularheptadecagon,thestepsshownin colorsequence; (2010):15 01,20 01,13 01,12 01,16 01,08 01,18A05, ,ISBN-13978-0-8176-3248-9c auserBostonDigitalSecondEdition,nottobes old,noISBNc ,images,andotherdatacontainedinthisfile, whichisinportabledocumentformat(PDF),are proprietarytotheauthor,andtheauthorretai nsallrights,includingcopyright, ,trademarks,servicemarks,andsimilaritems ,eveniftheyarenotidentifiedassuch, userBoston,c/oSpringerScience+BusinessMe diaInc.,233 SpringStreet,NewYork,NY10013,USA, ,scholarship,andresearch,andforthesepurp osesonly, ,postitonline,andtransmititdigitallyforp urposesofeducation,scholarship, ( ,EPUB),theymaynoteditit, ,usersmustchargenofee, ,noextractsorquotationsfromthisfilemaybe usedthatdonotconsistofwholepagesunlesspe rmissionhasbeengrantedbytheauthor(andbyB irkh userBostonifappropriate).

2 Thepermissiongrantedforuseofthewholefile andtheprohibitionagainstchargingfeesexte ndtoanypartialfilethatcontainsonlywholep agesfromthisfile, (andbyBirkh userBostonifappropriate).Inquiriesconcer ningprintcopiesofeithereditionshouldbedi rectedtoSpringerScience+ ,SarahandWilliam,andToMyAlgebraTeachers: RalphFox,JohnFraleigh,RobertGunning,John Kemeny,BertramKostant,RobertLanglands,Go roShimura,HaleTrotter, , +1and4n+ , (Continued) ,Polynomials, ,R, ,manyreadershavereactedtothebooksbysendi ngcomments,sugges-tions, , , ,theauthorgrantedapublishinglicensetoBir kh auserBostonthatwaslimitedtoprintmedia, ,andforeachbookaPDFfile,calledthe digitalsecondedition, ,thecorrectionofafewmisprints,onesmallam endmenttothe GuidefortheReader aboutChapterVII,someupdatestotheReferenc es, ,about2003,forBirkh , ,Birkh ausermathematicseditorinNewYorkasof2014, , , , ,Iplantopointtolistsofknowncorrectionsfr ommyownWebpage, ,whetherpureorapplied, ,itsuse, , ,includingnearlyallofBasicAlgebraandsome ofAdvancedAlgebra,correspondstonormalcou rsework.

3 Topicsbuildonthelinearalgebra,grouptheor y,factorizationofideals,struc-tureoffiel ds,Galoistheory,andelementarytheoryofmod ulesdevelopedinBasicAlgebra. Individualchapterstreatvarioustopicsinco mmutativeandnoncommutativealgebra,togeth erprovidingintroductionstothetheoryofass ociativealgebras,homologicalalgebra,alge braicnumbertheory,andalgebraicgeometry. Thetextemphasizesconnectionsbetweenalgeb raandotherbranchesofmath-ematics, ,itcarriesalongtwothemesfromBasicAlgebra :theanalogybetweenintegersandpolynomials inonevariableoverafield,andtherelationsh ipbetweennumbertheoryandgeometry. Severalsectionsintwochaptersintroducethe subjectofGr obnerbases,whichisthemoderngatewaytoward handlingsimultaneouspolynomialequationsi napplications. Thedevelopmentproceedsfromtheparticulart othegeneral, Morethan250problemsattheendsofchaptersil luminateaspectsofthetext,developrelatedt opics, HintsforSolutionsofProblems attheendofthebookgivesdetailedhintsformo stoftheproblems, ,grouptheory,ringsandmodules,uniquefacto rizationdomains,Dedekinddomains,fieldsan dalgebraicextensionfields, , , ,onenumber-theoryproblemalreadysolvedbyF ermatandEulerwastofindallpairs(x,y)ofint egerssatisfyingx2+y2=n, ,suchasy2=x3+.

4 Ifweregardeachequationasanexpressionsete qualto0, , ,particularlyinanalgebraicallyclosedfiel d, , ,knownastheBrauergroup, ,homologyandcohomologycanbeabstractedins uchawaythatthetheoryimpactsseveralfields simultaneously, ,algebraicgeometry, ,makesuseoftoolsfromanalysisconcerningco mpactnessandcompleteness,succeedsingivin gfullproofsofthethreetheoremsofChapterV, fromthealgebraicpointofviewofsimultaneou ssystemsofpolynomialequationsinseveralva riables,fromthenumber-theoreticpointofvi ewsuggestedbytheclassicaltheoryofRiemann surfaces, ,homologicalalgebrainChapterIV, ,and,asmentionedabove, ,someblocksofproblemsformadditionaltopic sthatcouldhavebeenincludedinthetextbutwe renot;theseblocksmayberegardedasoptional topics, , ,someareexamplesshowingthedegreetowhichh ypothesescanbestretched, , , ,propositions,lemmas, , , PROOF or PROOF ismatchedbyanoccurrenceattherightmargino fthesymbol ,andIamindebtedtoDavidKramer, , , , ,notnecessarilycommutative,andonechapter treatshomologicalalgebra.

5 Boththesetopicsplayaroleinalgebraicnumbe rtheoryandalgebraicgeometry, , , ,Zorn sLemma, IVofBasicAlgebraotherthantheSylowTheorem s,factsfromChapterVaboutdeterminantsandc haracteristicpolynomialsandminimalpolyno mials,simplepropertiesofmultilinearforms fromChapterVI,thedefinitionsandelementar ypropertiesofidealsandmodulesfromChapter VIII,theChineseRemainderTheoremandthethe oryofuniquefactorizationdomainsfromChapt erVIII, ,somesectionsofthebook,asindicatedbelow, ,uniformconvergence,differentialcalculus inseveralvariables, ,includingtheCauchyIntegralFormula,expan sionsinconvergentpowerseries, ,Euler,andLagrangetothealgebraicnumberth eoryofKummer,Dedekind,Kronecker,Hermite, sLawofQuadraticReciprocity,thetheoryofbi naryquadraticformsbegunbyGaussandcontinu edbyDirichlet,andDirichlet , ,whichoccurredlaterhistorically,anditant icipatesthedefinitionoftheclassnumberofa nalgebraicnumberfield, , ,Sections6 7usefactsaboutquadraticnumberfields,incl udingthemultiplicationofidealsintheirrin gsofintegers,andSection10usestheFourieri nversionformulaforfiniteabeliangroups, , , 3ofChapterII,whichassumesfamiliaritywith commutativeNoetherianringsasinChapterVII IofBasicAlgebra,plusthematerialinChapter Xonsemisimplemodules,chainconditionsform odules,andtheJordan H 6containthestatementandproofofWedderburn sMainTheorem, sTheoremthateveryfinitedivisionringiscom -mutativeandFrobenius sTheoremthattheonlyfinite-dimensionalass ociativedivisionalgebrasoverRareR,C,andt healgebraHofquaternions, , , ,withemphasisonconnectinghomo-morphisms, longexactsequences, good categoriesofunitalleftRmodules,Rbeingari ngwithidentity, ; , ; good categories; sheaf abelian ,butsomesubstituteisstillpossible; ,alongwithknowledgeoftheingredientsofthe theory Noetherianrings,integralclosure, theDedekindDiscriminantTheorem,theDirich letUnitTheorem, ,usingnoadditionaltools, ,completions, ,andcompactnessplaysanimportantroleinSec tions9.

6 , 5assumexxiiGuidefortheReaderknowledgeofl ocalizations, sTheoreminSection5istiedtothenotionofsin gularities; ; , 4arerelativelyelementaryandconcerntheres ultantandpreliminaryformsofBezout 6concernintersectionmultiplicityforcurve sandmakeextensiveuseoflo-calizations;the goalisabetterformofBezout 10areindependentofSections5 6andintroducethetheoryofGr 3definedivisorsandthegenusofsuchacurve,w hileSections4 5provetheRiemann , 3areconcernedwithalgebraicsetsandtheirdi mension,Sections4 6treatmapsbetweenvarieties,andSections7 12areindependentofSections6 8anddotwothingssimultaneously:theytiethe theoreticalworkondimensiontothetheoryofG r obnerbasesinChapterVIII,makingdimensionc omputable, , membershipsymbol#Sor|S|numberofelementsi nS emptyset{x E|P}thesetofxinEsuchthatPholdsEccompleme ntofthesetEE F,E F,E Funion,intersection,differenceofsets E , E union,intersectionofthesetsE E F,E FcontainmentE F,E Fpropercontainment(a1,..,an)orderedn-tup le{a1,..,an}unorderedn-tuplef:E F,x f(x)function,effectoffunctionf gorfg,f Ecompositionofffollowingg,restrictiontoE f( ,y)thefunctionx f(x,y)f(E),f 1(E)directandinverseimageofasetinone-one correspondencematchedbyaone-oneontofunct ioncountablefiniteorinone-onecorresponde ncewithintegers2 AsetofallsubsetsofANumbersystems ijKroneckerdelta.

7 1ifi=j,0ifi =j nk binomialcoefficientnpositive,nnegativen> 0,n<0Z,Q,R,Cintegers,rationals,reals,com plexnumbersmax,minmaximum/minimumoffinit esubsetofreals[x]greatestinteger xifxisrealRez,Imzrealandimaginarypartsof complexz zcomplexconjugateofz|z|absolutevalueofzx xiiixxivNotationandTerminologyLinearalge braandelementarynumbertheoryFnspaceofn-d imensionalcolumnvectorsejjthstandardbasi svectorofFnV dualvectorspaceofvectorspaceVdimFVordimV dimensionofvectorspaceVoverfieldF0zerove ctor,matrix,orlinearmapping1orIidentitym atrixorlinearmappingAttransposeofAdetAde terminantofA[Mij]matrixwith(i,j)thentryM ij L matrixofLrelativetodomainorderedbasis andrangeorderedbasis x ydotproduct =isisomorphicto,isequivalenttoFpintegers moduloaprimep,asafieldGCDgreatestcommond ivisor iscongruentto Euler s functionGroups,rings,modules,andcategori es0additiveidentityinanabeliangroup1mult iplicativeidentityinagrouporring =isisomorphicto,isequivalenttoCmcyclicgr oupofordermunitinvertibleelementinringRw ithidentityR groupofunitsinringRwithidentityRnspaceof columnvectorswithentriesinringRRoopposit eringtoRwitha b=baMmn(R)m-by-nmatriceswithentriesinRMn (R)n-by-nmatriceswithentriesinRunitallef tRmoduleleftRmoduleMwith1m=mforallm MHomR(M,N)groupofRhomomorphismsfromMinto NEndR(M)ringofRhomomorphismsfromMintoMke r ,image kernelandimageof Hn(G,N)nthcohomologyofgroupGwithcoeffici entsinabeliangroupNsimpleleftRmodulenonz erounitalleftRmodulewithnopropernonzeroR submodulessemisimpleleftRmodulesum(=dire ctsum)ofsimpleleftRmodulesObj(C)classofo bjectsforcategoryCMorphC(A,B)

8 SetofmorphismsfromobjectAtoobjectBNotati onandTerminologyxxvGroups,rings,modules, andcategories,continued1 AidentitymorphismonACScategoryofS-tuples ofobjectsfromObj(C)productof{Xs}s S(X,{ps}s S)suchthatifAinObj(C)and{ s MorphC(A,Xs)}aregiven,thenthereexistsaun ique MorphC(A,X)withps = sforallscoproductof{Xs}s S(X,{is}s S)suchthatifAinObj(C)and{ s MorphC(Xs,A)}aregiven,thenthereexistsaun ique MorphC(X,A)with is= sforallsCoppcategoryoppositetoCCommutati veringsRwithidentityandfactorizationofel ementsidentitydenotedby1,allowedtoequal0 idealI=(r1,..,rn)idealgeneratedbyr1,..,r nprimeidealIproperidealwithab Iimplyinga Iorb IintegraldomainRwithnozerodivisorsandwit h1 =0R/IwithIprimealwaysanintegraldomainGL( n,R)groupofinvertiblen-by-nmatrices,entr iesinRChineseRemainderTheoremI1,..,Ingiv enidealswithIi+Ij=Rfori = :R nj=1R/IjyieldsisomorphismR nj=1Ij =R/I1 nj=1Ij=I1 sLemmaIfIisanidealcontainedinallmaximali dealsandMisafinitelygeneratedunitalRmodu lewithIM=M,thenM= A , [X1.]

9 ,Xn]polynomialalgebraoverRwithnindetermi natesR[x1,..,xn]Ralgebrageneratedbyx1,.. ,xnirreducibleelementr =0r/ R suchthatr=abimpliesa R orb R primeelementr =0r/ R suchthatwheneverrdividesab, ;inanyuniquefactorizationdomain,irreduci bleimpliesprimeGCDgreatestcommondivisori nuniquefactorizationdomainxxviNotationan dTerminologyFieldsFqafinitefieldwithq=pn elements,pprimeK/FanextensionfieldKofafi eldF[K:F]degreeofextensionK/F, ,dimFKK(X1,..,Xn)fieldoffractionsofK[X1, ..,Xn]K(x1,..,xn)fieldgeneratedbyKandx1, ..,xnnumberfieldfinite-dimensionalfielde xtensionofQGal(K/F)Galoisgroup,automorph ismsofKfixingFNK/F( )andTrK/F( )normandtracefunctionsfromKtoFToolsforal gebraicnumbertheoryandalgebraicgeometryN oetherianRcommutativeringwithidentitywho seidealssatisfytheascendingchainconditio n; [X]NoetherianIntegralclosureSituation:R= integraldomain,F=fieldoffractions,K/F= KintegraloverRxisarootofamonicpolynomial inR[X]integralclosureofRinKsetofx KintegraloverR,isaringRintegrallyclosedR equalsitsintegralclosureinFLocalizationS ituation:R=commutativeringwithidentity,S = 1 Rlocalization,pairs(r,s)withr Rands S,modulo(r,s) (r ,s )ift(rs sr )=0forsomet SpropertyofS 1RI S 1 Iisone-onefromsetofidealsIinRofformI=R JontosetofidealsinS 1 Rlocalringcommutativeringwithidentityhav ingauniquemaximalidealRPforprimeidealPlo calizationwithS=complementofPinRDedekind domainNoetherianintegrallyclosedintegral domaininwhicheverynonzeroprimeidealismax imal,hasuniquefactorizationofnonzeroidea lsasproductofprimeidealsDedekinddomainex tensionRDedekind,Ffieldoffractions,K/Ffi niteseparableextension, ,andanynonzeroprimeideal inRhas R= gi=1 PeiifordistinctprimeidealsPiwithPi R=.

10 Thesehave gi=1eifi=[K:F],wherefi=[T/Pi:R/ ]. sLawofQuadraticReciprocity,thetheoryofbi naryquadraticforms,andDirichlet ,whichisareductionformulaprovidingarapid methodfordecidingsolvabilityofcongruence sx2 5developthetheoryofbinaryquadraticformsa x2+bxy+cy2,wherea,b, , ,conditionsfortherepresentabilityofprime sbyformsofagivendiscriminant,canonicalre presenta-tivesofthefinitelymanyproperequ ivalenceclassesofagivendiscriminant,agro uplawforproperequivalenceclassesofformso fthesamediscriminantthatrespectsrepresen tabilityofintegersbytheclasses, 10concernDirichlet sproductformulafor n=1n sandshowshowEulerwasabletomodifyittoprov ethatthereareinfinitelymanyprimes4k+1and infinitelymanyprimes4k+ ,andSection10containstheproofofDirichlet ,andSections9 ,Diophantus,Fermat,Euler,Lagrange, , , swork, largelytheworkofGaussandDirichlet , +bxy+cy2=mforunknownintegersxandy;x21+x2 2+x23+x24=mforunknownintegersx1,x2,x3,x4 ;y2=x(x 1)(x+1)forunknownintegersxandy;andxn+yn= znforunknownintegersx,y, +y2=p,wherepisaprime, +y2 ,sinceotherwiseywouldbedivisiblebyp,x2an dy2wouldbedivisiblebyp2,andx2+y2=pwouldb edivisiblebyp2, ,obtaining1+(yx 1)2 1modpforz xy ,then 1hasorder2,andthenecessaryconditionistha tthereexistsomezinF piscyclicoforderp 1,thenecessaryconditionisthat4dividep ,wecanestablishconverselythatthedivisibi lityofp 1by4impliesthatx2+y2= ,weknowfromthesolvabilityofz2 1modpthatthereexistsanintegerrsuchthatpd ividesr2+ [i]ofGaussianintegers,wherei= [i] [i]isaprincipalidealdomain, [i],thenthefactthatpdivides(r+i)(r i)impliesthatpdividesr+iorr iinZ[i].