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I . Algebra f ‘ Second :Edition' ' Z 2'. n. herstein Uniuzm'gya]Chicago TOPICS IN ALGEBRA 2nd edition JOHNWILEY&SONS NewYork - Chichester- Brisbane 'Toronto ToMarianne Copyright©1975,1964byXeroxCorporation. Allrightsreserved. Reproductionortranslationofanypanofthisworkbeyond thatperminedbySections107or108oflhe1976UniiedStates CopyrightActwithoutthepermissionofKhscopyrightowner isunlawiul‘RequestsForpermissionoriunherinformation shouldbeaddressedi0"isPermissionsDepartmentJohn Wiley5.Sons.inc. mrymuongmsCatalogCardNumber:74432577 PrimedintheUnitedStatesofAmerica. l09 Preface to the SecondEdition I approached revising Topic: in Algebra with a certain amount of trepidation. Onthewhole,Iwassatisfiedwiththefirsteditionanddid notwanttotamperwithit. However,therewerecertainchangesIfelt shouldbemade,changeswhichwouldnotaffectdiegeneralstyleor content, butwhichwouldmakethebookalittlemorecomplete. I hopethatIhaveachievedthisobjectiveinthepresentversion. Forthemostpart,themajorchangestakeplaceinthechapteron group theory. When the firsteditionwaswrittenitwasfairlyun- common for a student learning abstract algebra to have had any previousexposuretolinearalgebra. Nowadaysquitetheoppositeis true;manystudents,perhapsevenamajority,havelamedsomething about2 x 2matricesatthisstage. ThusIfeltfreeheretodrawon 2 x 2 matrices for examples and problems, These parts, which dependonsomeknowledgeoflinearalgebra,areindicatedwitha#. InthechapterongroupsIhavelargelyexpandedonesection,that onSylow’stheorem,andaddedtwoothers,oneondirectproductsand oneonthestructureoffiniteabeliangroups. IntheprevioustreatmentofSylow’stheorem,onlytheexistenceofa Sylow subgroup wasshown. This wasdonefollowing the proofof Wielandt. TheconjugacyoftheSylowsubgroupsandtheirnumber weredeveloped inaseriesofexercises, but notinthe textproper. NowallthepartsofSylow’stheoremaredoneinthetextmaterial. PrefacetotheSecondEdition In addition to the proofpreviously given for the existence, two other proofs of existence are carried out. One could accuse me ofoverkill atthispoint,probablyrightfullyso. ThefactofthematteristhatSylow’s theoremisimportant,thateachproofillustratesadifferentaspectofgroup theoryand,aboveall,thatIloveSylow’stheorem. Theproofofthecon- jugacyandnumberofSylowsubgroupsexploitsdoublecosets. Aby-product ofthisdevelopmentisthatameansisgivenforfindingSylowsubgroupsina largesetofsymmetricgroups. Forsomemysteriousreasonknownonlytomyself,I hadomitteddirect products in the first edition. Whyis beyond me. The materialis easy, straightforward, and important. This lacuna is nowfilled in the section treatingdirectproducts. Withthisinhand,Igooninthenextsectionto prove the decompositionofafiniteabeliangroupasa directproductof cyclicgroupsandalsoprovetheuniquenessoftheinvariantsassociatedwith thisdecomposition. Inpointoffact,thisdecompositionwasalreadyinthe firstedition,attheendofthechapteronvectorspaces,asaconsequenceof thestructureoffinitelygeneratedmodulesoverEuclideanrings. However, thecaseofafinitegroupisofgreatimportancebyitself;thesectiononfinite abeliangroupsunderlinesthisimportance. Itspresenceinthechapteron groups,anearlychapter,makesitmorelikelythatitwillbetaught. Oneotherentiresectionhasbeenaddedattheendofthechapteronfield theory. Ifeltthatthestudentshouldseeanexplicitpolynomialoveran explicitfieldwhoseGaloisgroupwasthesymmetricgroupofdegree5,hence onewhoserootscouldnotbeexpressedbyradicals. Inordertodoso, a theoremis firstproved which gives a criterion that an irreducible poly- nomialofdegreep,paprime,overtherationalfieldhaveSpasitsGalois group. As an application ofthis criterion, an irreducible polynomial of degree5isgiven,overtherationalfield,whoseGaloisgroupisthesymmetric groupofdegree5. Thereareseveralotheradditions. Morethan150newproblemsaretobe foundhere. Theyareofvaryingdegreesofdifficulty. Manyareroutine and computational, many are very difficult Furthermore, some inter- polatoryremarksaremadeaboutproblemsthathavegivenreadersagreat dealofdifficulty. Someparagraphshavebeeninserted,othersrewritten,at placeswherethewritinghadpreviouslybeenobscureortooterse. Above I have described what I have added. What gave me greater difficultyabouttherevisionwas,perhaps,thatwhichIhavenotadded. I debatedforalongtimewithmyselfwhetherornottoaddachapteron categorytheoryandsomeelementaryfunctors,whetherornottoenlargethe materialonmodulessubstantially. Afteragreatdealofthoughtandsoul- searching,Idecidednottodoso. Thebook,asstands,hasacertainconcrete- nessaboutitwithwhichthisnewmaterialwouldnotblend. Itcouldbe madetoblend,butthiswouldrequireacompletereworkingofthematerial PrefacetotheSecondEdition ofthebookandacompletechangeinitsphilosophy—somethingIdidnot wanttodo. Amereadditionofthisnewmaterial,asanadjunctwithno applications and no discernible goals, would have violated myguiding principle that all matters discussed should lead to some clearly defined objectives,tosome—highlight,tosomeexcitingtheorems. ThusIdecidedto omittheadditionaltopics. Manypeoplewrotemeaboutthefirsteditionpointingouttypographical mistakesormakingsuggestionsonhowtoimprovethebook. Ishouldliketo takethisopportunitytothankthemfortheirhelpandkindness. Preface to the First Edition Theideatowritethisbook,andmoreimportantthedesiretodoso,is adirectoutgrowthofacourseIgaveintheacademicyear1959—1960at CornellUniversity. Theclasstakingthiscourseconsisted,inlargepart, ofthemostgiftedsophomoresinmathematicsatCornell. Itwasmy desiretoexperimentbypresentingtothemmaterialalittlebeyondthat whichisusuallytaughtinalgebraatthejunior-seniorlevel. Ihaveaimedthisbooktobe,bothincontentanddegreeofsophisti~ cation,abouthalfwaybetweentwogreatclassics,ASurveyufModern Algebra, byBirkhofl"and MacLane,andModernAlgebra,byVander Waerden. Thelastfewyearshaveseenmarkedchangesintheinstructiongiven in mathematics at the American universities. This change is most notable attheupperundergraduateand beginninggraduatelevels. Topicsthatafewyearsagowereconsideredpropersubjectmatterfor semiadvancedgraduatecoursesinalgebrahavefiltereddownto,and arebeingtaughtin,theveryfirstcourseinabstractalgebrar Convinced thatthisfiltrationwillcontinueandwillbecomeintensifiedinthenext fewyears,Ihaveputintothisbook,whichisdesignedtobeusedasthe student'sfirstintroductiontoalgebra,materialwhichhithertohasbeen consideredalittleadvancedforthatstageofthegame. Thereisalwaysagreatdangerwhentreatingabstractideastointro- ducethemtoosuddenlyandwithoutasufficientbaseofexamplesto renderthemcredibleornatural. Inordertotrytomitigatethis,Ihave triedtomotivatetheconceptsbeforehandandtoillustratethemincon- cretesituations.Oneofthemosttellingproofsoftheworthofanabstract PrefacetotheFirstEdition conceptiswhatit,andtheresultsaboutit,tellsusinfamiliarsituations. In almosteverychapteranattemptismadetobringoutthesignificanceofthe generalresultsbyapplyingthemtoparticularproblems. Forinstance,inthe chapteronrings,thetwo-squaretheoremofFermatisexhibitedasadirect consequenceofthetheorydevelopedforEuclideanrings. Thesubjectmatterchosenfordiscussionhasbeenpickednotonlybecause ithasbecomestandardtopresentitatthislevelorbecauseitisimportantin thewholegeneraldevelopmentbutalsowithaneyetothis“concreteness.” ForthisreasonIchosetoomittheJordan-Holdertheorem,whichcertainly couldhaveeasilybeenincludedintheresultsderivedaboutgroups. How- ever,toappreciatethisresultforitsownsakerequiresagreatdealofhind- sightandtoseeitusedeffectivelywouldrequiretoogreatadigression. True, onecoulddevelopthewholetheoryofdimensionofavectorspaceasoneof itscorollaries,but,forthefirsttimearound,thisseemslikeamuchtoofancy andunnaturalapproachtosomethingsobasicanddown-to-earth. Likewise, thereisnomentionoftensorproductsorrelatedconstructions. Thereisso much time and opportunity to become abstract; why rush it at the beginning? Awordabouttheproblems. Thereareagreatnumberofthem. Itwould bean extraordinarystudentindeedwhocouldsolve themall. Someare presentmerelytocompleteproofsin thetextmaterial,otherstoillustrate andtogivepracticein theresultsobtained. Manyareintroducednotso muchtobesolvedastobetackled. Thevalueofaproblemisnotsomuch incomingupwiththeanswerasintheideasandattemptedideasitforces onthewould-besolver. Othersareincludedinanticipationofmaterialto bedevelopedlater,thehopeand rationaleforthisbeingbothtolaythe groundworkforthesubsequenttheoryandalsotomakemorenaturalideas, definitions,andargumentsastheyareintroduced. Severalproblemsappear morethanonce. Problemsthat{orsomereasonorotherseemdifficulttome areoftenstarred(sometimeswithtwostars). However,evenheretherewill benoagreementamongmathematicians;manywillfeelthatsomeunstarred problemsshouldbestarredandviceversa. Naturally,Iamindebtedtomanypeopleforsuggestions,commentsand criticisms. Tomentionjustafewofthese:CharlesCurtis,MarshallHall, NathanJacobson,ArthurMattuck,andMaxwellRosenlicht. Ioweagreat deal to Daniel Gorenstein and IrvingKaplanskyfor the numerous con- versations we have had about the book, its material and its approach. Aboveall,IthankGeorgeSeligmanforthemanyincisivesuggestionsand remarksthathehasmadeaboutthepresentationbothastoitsstyleandto itscontent. IamalsogratefultoFrancisMcNaryofthestafi‘ofGinnand Companyforhishelpandcooperation. Finally,Ishouldliketoexpressmy thankstotheJohnSimonGuggenheimMemorialFoundation;thisbookwas in part written with their support while the author was in Rome as a GuggenheimFellow.