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UNITEXT 140 Rocco Chirivì · Ilaria Del Corso Roberto Dvornicich Selected Exercises in Algebra Volume 2 UNITEXT La Matematica per il 3+2 Volume 140 Editor-in-Chief AlfioQuarteroni,PolitecnicodiMilano,Milan,Italy;ÉcolePolytechniqueFédérale deLausanne(EPFL),Lausanne,Switzerland SeriesEditors LuigiAmbrosio,ScuolaNormaleSuperiore,Pisa,Italy PaoloBiscari,PolitecnicodiMilano,Milan,Italy CiroCiliberto,UniversitédiRoma“TorVergata”,Rome,Italy CamilloDeLellis,InstituteforAdvancedStudy,Princeton,NJ,USA MassimilianoGubinelli,HausdorffCenter forMathematics,RheinischeFriedrich- Wilhelms-Universität,Bonn,Germany VictorPanaretos,InstituteofMathematics,ÉcolePolytechniqueFédéralede Lausanne(EPFL),Lausanne,Switzerland LorenzoRosasco,DIBRIS,UniversitédegliStudidiGenova,Genova,Italy;Center forBrainsMindandMachines,MassachusettsInstituteofTechnology,Cambridge, Massachusetts,US;IstitutoItalianodiTecnologia,Genova,Italy TheUNITEXT - La Matematicaper il3+2seriesis designedforundergraduate andgraduateacademiccourses,andalsoincludesadvancedtextbooksataresearch level. Originally released in Italian, the series now publishes textbooks in English addressedtostudentsinmathematicsworldwide. Some of the most successful books in the series have evolved through several editions,adaptingtotheevolutionofteachingcurricula. Submissions must include at least 3 sample chapters, a table of contents, and a preface outlining the aims and scope of the book, how the book fits in with the currentliterature,andwhichcoursesthebookissuitablefor. For any further information, please contact the Editor at Springer:francesca. [email protected] THESERIESISINDEXEDINSCOPUS *** UNITEXT is glad to announce a new series of free webinars and interviews handled by the Board members, who rotate in order to interview top experts in their field. Access this link to subscribe to the events: https://cassyni.com/events/ TPQ2UgkCbJvvz5QbkcWXo3 Rocco Chirivì • Ilaria Del Corso • Roberto Dvornicich Selected Exercises in Algebra Volume 2 RoccoChirivì IlariaDelCorso DepartmentofMathematicsandPhysics DepartmentofMathematics UniversityofSalento UniversityofPisa Lecce,Italy Pisa,Italy RobertoDvornicich DepartmentofMathematics UniversityofPisa Pisa,Italy Translatedby AlessandraCaraceni ScuolaNormaleSuperiore Pisa,Italy ISSN2038-5714 ISSN2532-3318 (electronic) UNITEXT ISSN2038-5722 ISSN2038-5757 (electronic) LaMatematicaperil3+2 ISBN978-3-031-12650-5 ISBN978-3-031-12651-2 (eBook) https://doi.org/10.1007/978-3-031-12651-2 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Coverillustration:“Squares,circlesandsymmetries2”,RoccoChirivì(2018) ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland “Thenicethingaboutmathematicsis doing mathematics.” —PierreDeligne To Andrea,who knows what mathematicsis Rocco To Francesca,with awish thatshewillbe ableto findandnurtureher passions Ilaria To youngpeoplewho alreadyloveormight cometolovemathematics Roberto Preface This second volume of selected exercises is meant as a companion to the first. Like its predecessor, it contains exam questions set in recent years in the context of the bachelor’s degree in mathematics at the University of Pisa. The problems areaccompaniedbyfullsolutions,abriefoverviewoftheunderlyingtheory,anda seriesofpreliminaryexercises. Ourmotivationinwritingthisbookandtheobjectiveswehopetoachievewithit remaintheonesdescribedintheprefacetothefirstvolume.Verybriefly,thesebooks stemfromourconvictionthat,inordertostudyanddeeplyunderstandalgebra(and, indeed,mathematicsingeneral),attendinglecturesandmemorisingthestatements andproofsofkeyresultsarenotenough.Whatoneneedstodoisapplythetheory onestudiestoconcreteexamples;inpractice,useittosolveproblems. This being said, problems come in different varieties. Some require the appli- cation of rather simple procedures, or their solution can be derived immediately fromdefinitionsandtheorem;othersdonotonlyrequiresufficientknowledgeofthe theory,butalsosomenovelinsightbymeansofwhichnewinformationisunlocked. The reader will soon realise that problems of the first type are conspicuously absentfrom this book;they will be forcedto abandonthe hope for easy solutions and instead engage with the material at a much deeper level. But this is how it’s done: mathematics is not a novel to be passively read, but a story to be actively reinvented. Thisiswhyweheartilyrecommendthatthereaderarmthemselveswithpatience anddonotlookupthesolutiontoanexercisebeforetheyhavespentaconsiderable amountoftimetryingtocomeupwithitthemselves. Thereareacouplemoreideasthatwewouldliketostress.Thefirstisthatmaths enthusiastsatalllevelsaretakeninbythebeautyofmathematics;thesecondisthat beingenthusiasticaboutanythinginvolvescuriosityforthetruth.We havetriedto embodywithinmanyoftheexercisesinthisbookourpersonalsenseofbeautyand thespiritofcuriositythathasalwaysinspiredus.Wehopethatthereadermightalso benefitfromthisaspectofourwork. Asisthecaseforthefirstvolume,thebookisstructuredaccordingtothewaythat algebraistaughtduringthefirstyearsofamathematicsdegreeattheUniversityof ix x Preface Pisa.Whenthedistinctionbetweenbachelor’sandmaster’sdegreeswasintroduced, whatusedtobeasinglealgebracoursewassplitintotwoparts,whichareatpresent labelled“Arithmetic”and“Algebra1”;thesetwopartsperfectlycorrespondtothe twovolumesofthiswork. The “Arithmetic” portion of the material, to which the first volume is devoted, ismostlyconcernedwithbasictoolssuchasinduction,someelementsofcombina- torics,integersandcongruences,andincludesanintroductiontothebasicproperties of fundamentalalgebraic structures: Abelian groups, rings, polynomials and their roots,fieldextensionsandfinitefields. The“Algebra1”portioncoveredbythissecondvolumedelvesdeeperintogroup theory,discussescommutativeringswithaparticularfocusonuniquefactorisation, anddealswithfieldextensionsandthefundamentalsofGaloistheory. Each part is accompanied by notes on the theory. Although exhaustive, they do not attempt to make up for an algebra textbook, as evidenced by the fact that they contain the statements of relevant results but provide no proofs. (For further information the reader may instead turn, for example, to Herstein’s “Topics in Algebra”,Wiley&Sons,orArtin’s“Algebra”,Pearson.) Thisbookalsoincludesaseriesofpreliminaryexercises.Theseshouldbetackled first,sincetheyyieldresultsthatwilloftenbeneededinsubsequentproblems.We alsostressthatnoneofthesolutionsprovidedrequireanytheoreticaltoolsotherthan thoserecalledinthe introductorychaptersandthe preliminaryexercises.Invoking moreadvancedtheoremswouldsimplifycertainsolutionsandoccasionallyrender anexercisetrivial,butdoingsowouldbecontrarytotheveryspiritofthiswork. Acknowledgements WewouldliketothankFilippoCallegaroforhiscollaboration in preparingsome of the exercises and Alessandro Berarducciand Ilaria Damiani fortheirpreciousadvice.WethankDr.FrancescaBonadeiandDr.FrancescaFerrari ofSpringerItaliafortheirassistance.Finally,aspecialthankstoallthestudentswho attendedourlecturesandbusiedthemselveswiththeexercisesduringexams. Updates We warmly invite readers to provide us with feedback and point out mistakes (which, in a book containing the detailed solutions to hundreds of exercises, are almost inevitably to be found) by contacting us at the addresses: [email protected] [email protected] [email protected]. Updatesandcorrectionswillappearonthewebpagehttp://www.dmf.unisalento. it/~chirivi/libroEserciziAlgebra.html. Lecce,Italy RoccoChirivì Pisa,Italy IlariaDelCorso Pisa,Italy RobertoDvornicich July2018 Contents 1 Theory......................................................................... 1 1.1 Groups .................................................................. 1 1.1.1 BasicNotions ................................................. 1 1.1.2 HomomorphismTheorems ................................... 3 1.1.3 FreeGroups ................................................... 4 1.1.4 PresentationsofGroups ...................................... 6 1.1.5 TheDihedralGroup .......................................... 7 1.1.6 GroupAutomorphisms ....................................... 8 1.1.7 CommutatorsandAbelianisation ............................ 9 1.1.8 GroupActions ................................................ 10 1.1.9 TheActionbyConjugation .................................. 12 1.1.10 TheActionbyMultiplication ................................ 13 1.1.11 p-Groups ...................................................... 14 1.1.12 Permutations .................................................. 15 1.1.13 AbelianGroups ............................................... 18 1.1.14 SemidirectProduct ........................................... 19 1.1.15 SylowTheorems .............................................. 20 1.1.16 SimpleGroups ................................................ 21 1.2 Rings .................................................................... 22 1.2.1 BasicNotions ................................................. 22 1.2.2 MaximalandPrimeIdeals ................................... 23 1.2.3 QuotientRings ................................................ 24 1.2.4 IdealOperations .............................................. 24 1.2.5 FractionFieldsandLocalisations ............................ 26 1.2.6 Divisibility .................................................... 27 1.2.7 EuclideanDomains ........................................... 29 1.2.8 PrincipalIdealDomains ...................................... 31 1.2.9 UniqueFactorisationDomains ............................... 32 1.2.10 GaussianIntegers ............................................. 35 xi