Glimpses of Algebra and Geometry, Second Edition Gabor Toth Springer Undergraduate Texts in Mathematics Readings in Mathematics Editors S. Axler F.W. Gehring K.A. Ribet Gabor Toth Glimpses of Algebra and Geometry Second Edition With 183 Illustrations, Including 18 in Full Color GaborToth DepartmentofMathematicalSciences RutgersUniversity Camden,NJ08102 USA [email protected] EditorialBoard S.Axler F.W.Gehring K.A.Ribet MathematicsDepartment MathematicsDepartment MathematicsDepartment SanFranciscoState EastHall UniversityofCalifornia, University UniversityofMichigan Berkeley SanFrancisco,CA94132 AnnArbor,MI48109 Berkeley,CA94720-3840 USA USA USA Front cover illustration: The regular compound of five tetrahedra given by the face- planes of a colored icosahedron. The circumscribed dodecahedron is also shown. Computergraphic madeby theauthor usingGeomview. Backcoverillustration: The regularcompoundoffivecubesinscribedinadodecahedron.Computergraphicmade bytheauthorusingMathematica. MathematicsSubjectClassification(2000):15-01,11-01,51-01 LibraryofCongressCataloging-in-PublicationData Toth,Gabor,Ph.D. Glimpsesofalgebraandgeometry/GaborToth.—2nded. p.cm.—(Undergraduatetextsinmathematics.Readingsinmathematics.) Includesbibliographicalreferencesandindex. ISBN0-387-95345-0(hardcover:alk.paper) 1.Algebra. 2.Geometry. I.Title. II.Series. QA154.3.T68 2002 512′.12—dc21 2001049269 Printedonacid-freepaper. 2002,1998Springer-VerlagNewYork,Inc. All rights reserved. This work may not be translated or copied in whole or in part withoutthewrittenpermissionofthepublisher(Springer-VerlagNewYork,Inc.,175 FifthAvenue,NewYork,NY10010,USA),exceptforbriefexcerptsinconnectionwith reviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstor- ageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimi- larmethodologynowknownorhereafterdevelopedisforbidden. The use in this publication of trade names, trademarks, service marks, and similar terms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionof opinionastowhetherornottheyaresubjecttoproprietaryrights. ProductionmanagedbyFrancineMcNeill;manufacturingsupervisedbyJeffreyTaub. Typeset from the author’s 2e files using Springer’s UTM style macro by The BartlettPress,Inc.,Marietta,GA. PrintedandboundbyHamiltonPrintingCo.,Rensselaer,NY. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 ISBN0-387-95345-0 SPIN10848701 Springer-Verlag NewYork Berlin Heidelberg AmemberofBertelsmannSpringerScience+BusinessMediaGmbH This book is dedicated to my students. Springer-VerlagElectronicProduction toth 12:27p.m.2·v·2002 ........................................... PSerceofancde Etoditthioen Since the publication of the Glimpses in 1998, I spent a consider- able amount of time collecting “mathematical pearls” suitable to addtotheoriginaltext.Asmycollectiongrew,itbecameclearthat a major revision in a second edition needed to be considered. In addition,manyreadersoftheGlimpsessuggestedchanges,clarifi- cations, and, above all, more examples and worked-out problems. This second edition, made possible by the ever-patient staff of Springer-Verlag New York, Inc., is the result of these efforts. Al- though the general plan of the book is unchanged, the abundance oftopicsrichinsubtleconnectionsbetweenalgebraandgeometry compelled me to extend the text of the first edition considerably. Throughouttherevision,Itriedtodomybesttoavoidtheinclusion of topics that involve very difficult ideas. The major changes in the second edition are as follows: 1. Anin-depthtreatmentofrootformulassolvingquadratic,cubic, andquarticequationsa` lavanderWaerdenhasbeengivenina new section. This can be read independently or as preparation for the more advanced new material encountered toward the later parts of the text. In addition to the Bridge card symbols, the dagger † has been introduced to indicate more technical material than the average text. vii Springer-VerlagElectronicProduction toth 12:27p.m.2·v·2002 viii PrefacetotheSecondEdition 2. AsanaturalcontinuationofthesectiononthePlatonicsolids,a detailedandcompleteclassificationoffiniteMo¨biusgroupsa` la Klein has been given with the necessary background material, such as Cayley’s theorem and the Riemann–Hurwitz relation. 3. Oneofthemostspectaculardevelopmentsinalgebraandgeom- etryduringthelatenineteenthcenturywasFelixKlein’stheory oftheicosahedronandhissolutionoftheirreduciblequinticin termsofhypergeometricfunctions.Aquick,direct,andmodern approach of Klein’s main result, the so-called Normalformsatz, hasbeengiveninasinglelargesection.Thistreatmentisinde- pendent of the material in the rest of the book, and is suitable for enrichment and undergraduate/graduate research projects. All known approaches to the solution of the irreducible quin- tic are technical; I have chosen a geometric approach based on theconstructionofcanonicalquinticresolventsoftheequation of the icosahedron, since it meshes well with the treatment of the Platonic solids given in the earlier part of the text. An al- gebraic approach based on the reduction of the equation of the icosahedrontotheBrioschiquinticbyTschirnhaustransforma- tions is well documented in other textbooks. Another section onpolynomialinvariantsoffiniteMo¨biusgroups,andtwonew appendices, containing preparatory material on the hyperge- ometric differential equation and Galois theory, facilitate the understanding of this advanced material. 4. The text has been upgraded in many places; for example, there is more material on the congruent number problem, the stereographicprojection,theWeierstrass℘-function,projective spaces, and isometries in space. 5. The new Web site at http://mathsgi01.rutgers.edu/∼gtoth/ Glimpses/containingvarioustextfiles(inPostScriptandHTML formats) and over 70 pictures in full color (in gif format) has been created. 6. The historical background at many places of the text has been made more detailed (such as the ancient Greek approxima- tionsofπ),andthehistoricalreferenceshavebeenmademore precise. 7. Anextendedsolutionsmanualhasbeencreatedcontainingthe solutions of 100 problems. Springer-VerlagElectronicProduction toth 12:27p.m.2·v·2002 PrefacetotheSecondEdition ix Iwouldliketothankthemanyreaderswhosuggestedimprove- ments to the text of the first edition. These changes have all been incorporated into this second edition. I am especially indebted to Hillel Gauchman and Martin Karel, good friends and colleagues, who suggested many worthwhile changes. I would also like to ex- press my gratitude to Yukihiro Kanie for his careful reading of the text and for his excellent translation of the first edition of the Glimpses into Japanese, published in early 2000 by Springer- Verlag, Tokyo. I am also indebted to April De Vera, who upgraded thelistof Websitesinthefirstedition.Finally,Iwouldliketothank InaLindemann,ExecutiveEditor,Mathematics,atSpringer-Verlag New York, Inc., for her enthusiasm and encouragement through- out the entire project, and for her support for this early second edition. Camden, New Jersey Gabor Toth Springer-VerlagElectronicProduction toth 12:27p.m.2·v·2002 ........................................... PFirresftaEcedittoiotnhe Glimpse: 1.averybriefpassing look,sightorview. 2.amomentary orslightappearance. 3.avague ideaorinkling. —RandomHouseCollegeDictionary At the beginning of fall 1995, during a conversation with my re- spected friend and colleague Howard Jacobowitz in the Octagon Dining Room (Rutgers University, Camden Campus), the idea emerged of a “bridge course” that would facilitate the transition between undergraduate and graduate studies. It was clear that a course like this could not concentrate on a single topic, but shouldbrowsethroughanumberofmathematicaldisciplines.The selectionoftopicsfortheGlimpsesthusprovedtobeofutmostim- portance.Atthislevel,themostprominentinterplayismanifested in some easily explainable, but eventually subtle, connections be- tween number theory, classical geometries, and modern algebra. Therich,fascinating,andsometimespuzzlinginteractionsofthese mathematical disciplines are seldom contained in a medium-size undergraduatetextbook.TheGlimpsesthatfollowmakeahumble effort to fill this gap. xi Springer-VerlagElectronicProduction toth 12:27p.m.2·v·2002 xii PrefacetotheFirstEdition The connections among the disciplines occur at various levels inthetext.Theyaresometimesthemaintopics,suchasRational- ity and Elliptic Curves (Section 3), and are sometimes hidden in problems,suchasthesphericalgeometricproofofdiagonalization ofEuclideanisometries(Problems1to2,Section16),ortheproof ofEuler’stheoremonconvexpolyhedrausinglinearalgebra(Prob- lem9,Section20).Despitenumerousopportunitiesthroughoutthe text,theexperiencedreaderwillnodoubtnoticethatanalysishad to be left out or reduced to a minimum. In fact, a major source of difficulties in the intense 8-week period during which I pro- ducedthefirstversionofthetextwasthecontinuouscuttingdown of the size of sections and the shortening of arguments. Further- more, when one is comparing geometric and algebraic proofs, the geometricargument,thoughoftenmorelengthy,isalmostalways morerevealingandtherebypreferable.Tostriveforsomeoriginal- ity, I occasionally supplied proofs out of the ordinary, even at the “expense” of going into calculus a bit. To me, “bridge course” also meant trying to shed light on some of the links between the first recordedintellectualattemptstosolveancientproblemsofnumber theory, geometry, and twentieth-century mathematics. Ignoring detours and sidetracks, the careful reader will see the continuity ofthelinesofarguments,someofwhichhaveatimespanof3000 years.Inkeepingthiscontinuity,Ieventuallydecidednottobreak up the Glimpses into chapters as one usually does with a text of this size. The text is, nevertheless, broken up into subtexts corre- sponding to various levels of knowledge the reader possesses. I havechosenthecardsymbols♣,♦,♥,♠ofBridgetoindicatefour levels that roughly correspond to the following: ♣ College Algebra; ♦ Calculus, Linear Algebra; ♥NumberTheory,ModernAlgebra(elementarylevel),Geometry; ♠ModernAlgebra(advancedlevel),Topology,ComplexVariables. Althoughmuchof♥and♠canbeskippedatfirstreading,Iencour- age the reader to challenge him/herself to venture occasionally intotheseterritories.Thebookisintendedfor(1)students(♣and ♦) who wish to learn that mathematics is more than a set of tools (thewaysometimescalculusistaught),(2)students(♥and♠)who