Basic Algebra - McGill University

1 Basic ,withanappendix ElementaryComplexAnalysis AdvancedRealAnalysisAnthony W. KnappBasic AlgebraAlong with a Companion VolumeAdvanced 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.

2 ,233 SpringStreet,NewYork,NY10013,USA, ,scholarship,andresearch,andforthesepurp osesonly, ,postitonline,andtransmititdigitallyforp urposesofeducation,scholarship, ( ,EPUB),theymaynoteditit, ,usersmustchargenofee, ,noextractsorquotationsfromthisfilemaybe usedthatdonotconsistofwholepagesunlesspe rmissionhasbeengrantedbytheauthor(andbyB irkh userBostonifappropriate).Thepermissiongr antedforuseofthewholefileandtheprohibiti onagainstchargingfeesextendtoanypartialf ilethatcontainsonlywholepagesfromthisfil e, (andbyBirkh userBostonifappropriate).

3 Inquiriesconcerningprintcopiesofeithered itionshouldbedirectedtoSpringerScience+ ,SarahandWilliam,andToMyAlgebraTeachers: RalphFox,JohnFraleigh,RobertGunning,John Kemeny,BertramKostant,RobertLanglands,Go roShimura,HaleTrotter, ,POLYNOMIALS, ,R, ,LinearIndependence, (Continued) ,manyreadershavereactedtothebookbysendin gcomments,suggestions, , , ,andtherewereenoughcorrections,perhapsah undredinall, , , ,theauthorgrantedapublishinglicensetoBir kh auserBostonthatwaslimitedtoprintmedia, ,andaPDFfile,calledthe digitalsecondedition, : sLemma,theearlierproofhavinghadagap.

4 Anumberofproblemshavebeenaddedattheendso fthechapters, :(a)(ChapterII)therelationshipintwoandth reedimensionsbetweendeter-minantsandarea sorvolumes,(b)(ChaptersVandIX)furtherasp ectsofcanonicalformsformatricesandlinear mappings,(c)(ChapterVIII)amplificationof usesoftheFundamentalTheoremofFinitelyGen eratedModulesoverprincipalidealdomains,x ixiiPrefacetotheSecondEdition(d)(Chapter IX)theinterplayofextensionofscalarsandGa loistheory,(e)(ChapterIX)propertiesandex amplesoforderedfieldsandrealclosedfields . Somerevisionshavebeenmadetothechapteronf ieldtheory(ChapterIX).

5 Itwasoriginallyexpected,anditcontinuesto beexpected, ,theoriginalplacementofthebreakbetweenvo lumesleftsomepossibleconfusionaboutthero leof normalextensions infieldtheory,andthatmatterhasnowbeenres olved. Characteristicpolynomialsinitiallyhaveav ariable , throughoutmostofthebookintheoriginaledit ion, fromChapterVon,andcharacteristicpolynomi alshavebeentreatedunambiguouslythereafte rasabstractpolynomials. :theanalogybetweenintegersandpolynomials inonevariableoverafield,theinterplaybetw eenlinearalgebraandgrouptheory, ;anexampleofsuchanotionis universalmappingproperty.

6 Readerswillbenefitfromlookingfortheseand othersuchthemes, ,Birkh ausermathematicseditorinNewYork,whoencou ragedthewritingofasecondedition,whomadea numberofsuggestionsaboutpursuingit, , , , ,Iplantopointtoalistofknowncorrectionsfr ommyownWebpage, ,whetherpureorapplied, ,itsuse, , ,particularlyinBasicAlgebrabutalsoinsome ofthechaptersofAdvancedAlgebra, : ,asthetopiclikelytobebetterknownbytherea deraheadoftime,andthenalittlegrouptheory isintroduced,withlinearalgebraprovidingi mportantexamples. Chaptersonlinearalgebradevelopnotionsrel atedtovectorspaces,thetheoryoflineartran sformations,bilinearforms,classicallinea rgroups,andmultilinearalgebra.

7 Chaptersonmodernalgebratreatgroups,rings ,fields,modules,andGaloisgroups,includin gmanyusesofGaloisgroupsandmethodsofcompu tation. Threeprominentthemesrecurthroughoutandbl endtogetherattimes:theanalogybetweeninte gersandpolynomialsinonevariableoverafiel d,thein-terplaybetweenlinearalgebraandgr ouptheory,andtherelationshipbetweennumbe rtheoryandgeometry. Thedevelopmentproceedsfromtheparticulart othegeneral,oftenintroducingexampleswell beforeatheorythatincorporatesthem. Morethan400problemsattheendsofchaptersil luminateaspectsofthetext,developrelatedt opics, HintsforSolutionsofProblems attheendofthebookgivesdetailedhintsformo stoftheproblems,completesolutionsformany .

8 ApplicationssuchasthefastFouriertransfor m,thetheoryoflinearerror-correctingcodes ,theuseofJordancanonicalforminsolvinglin earsystemsofordinarydifferentialequation s, , , ,inthe1920sEmmyNoetherintroducedvectorsp acesandlinearmappingstoreinterpretcoordi natespacesandmatrices,andshedefinedthein gredientsofwhatwasthencalled modernalgebra theaxiomaticallydefinedrings,fields,andm odules, , ,withmanyapplications, ,cryptography,andadvancesinphysicsandche mistryhavechangedallthat, ,andtheytoohaveextensiveapplicationsinsc ienceandengineering, , ,rings,fields, ;ofthese,theWedderburntheoryofsemisimple algebras,homologicalalgebra, VIItreatlinearalgebraandgrouptheoryatvar iouslevels,exceptthatthreesectionsofChap terIVandoneofChapterVintroduceringsandfi elds,polynomials,categoriesandfunctors, ,withemphasisonuniquefactorization;Chapt erIXconcernsfieldextensionsandGaloistheo ry,withemphasisonapplicationsofGaloisthe ory.

9 VandpartsofChaptersVIIIandIX, ,itmaybepossibletoskimmuchofthefirstthre echaptersandsomeofthebeginningofthefourt h;thentimemayallowforsomeofChaptersVIand VII,oradditionalmaterialfromChaptersVIII andIX, , ,plusthefirstsixsectionsofChapterIVandas muchasreasonablefromChapterV; ;thecoursewillperhapstreattheremainderof ChapterIV,thefirstfiveorsixsectionsofCha pterVIII,andatleastSections1 GuidefortheReader onpagesxxi ,IhavebuiltonecoursearoundChaptersI III,Sections1 6ofChapterIV,allofChapterV, ,alittleofChapterVII,Sections1 6ofChapterVIII,andSections1 ,someblocksofproblemsformadditionaltopic sthatcouldhavebeenincludedinthetextbutwe renot.

10 Theseblocksmayeitherberegardedasoptional topics, ,VII, , ,someareexamplesshowingthedegreetowhichh ypothesescanbestretched, , , ,themostimportantprereq-uisiteforusingBa sicAlgebraisthatthereaderalreadyknowwhat aproofis,howtoreadaproof, ,orfromacourseinlinearalgebra,orfromafir stjunior ,itisassumedthatthereaderiscomfortablewi thasmallamountoflinearalgebra,includingm atrixcomputations,rowreductionofmatrices ,solutionsofsystemsoflinearequations, ,propositions,lemmas, ,andonecanfindthepagereferenceforeachfig urefromthetableonpagesxvii.