Abstract Algebra and Famous Impossibilities: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations PDF
Preview Abstract Algebra and Famous Impossibilities: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations
Undergraduate Texts in Mathematics Readings in Mathematics Sidney A. Morris Arthur Jones Kenneth R. Pearson Abstract Algebra and Famous Impossibilities Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations Second Edition Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics Readings in Mathematics Series Editors Pamela Gorkin Bucknell University, Lewisburg, PA, USA Jessica Sidman Mount Holyoke College, South Hadley, MA, USA Advisory Board Colin Adams, Williams College, Williamstown, MA, USA Jayadev S. Athreya, University of Washington, Seattle, WA, USA Nathan Kaplan, University of California, Irvine, CA, USA Lisette G. de Pillis, Harvey Mudd College, Claremont, CA, USA Jill Pipher, Brown University, Providence, RI, USA Jeremy Tyson, University of Illinois at Urbana-Champaign, Urbana, IL, USA Undergraduate Texts in Mathematics are generally aimed at third- and fourth-year undergraduate mathematics students at North American universities. Thesetextsstrivetoprovidestudentsandteacherswithnewperspectivesandnovel approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation ofinterrelations among different aspectsofthesubject.Theyfeature examples that illustrate key concepts as well as exercises that strengthen understanding. Sidney A. Morris (cid:1) Arthur Jones (cid:1) Kenneth R. Pearson Abstract Algebra and Famous Impossibilities Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations Second Edition With 28 Illustrations 123 SidneyA.Morris ArthurJones (Deceased) Department ofMathematical VIC,Australia andPhysical Sciences LaTrobe University Bundoora,VIC, Australia Schoolof Engineering ITandPhysical Sciences FederationUniversity Australia Ballarat, VIC,Australia Kenneth R.Pearson (Deceased) VIC,Australia ISSN 0172-6056 ISSN 2197-5604 (electronic) Undergraduate Texts inMathematics ISSN 2945-5839 ISSN 2945-5847 (electronic) Readingsin Mathematics ISBN978-3-031-05697-0 ISBN978-3-031-05698-7 (eBook) https://doi.org/10.1007/978-3-031-05698-7 MathematicsSubjectClassification: 12-01,12F05,12E05,01A55 ThefirsteditionofthistextbookwaspublishedintheseriesUniversitext. 1stedition:©Springer-VerlagNewYork,Inc.1991 2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringer NatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface to the Second Edition It is thirty years since the first edition of this book appeared. A distinguishing feature of the first edition and this edition is that the study of abstract algebra is motivated by demonstrating that it is exactly what was needed to solve several famous problems which had remained unsolved for over 2,000 years. In the first edition we showed how the three geometric problems known as Doubling the Cube, Trisecting an Angle, and Squaring the Circle were finally solved. In this edition, a famous fourth ancient problem, namely Solving Polynomial Equations, is added to this list. Inthelastchapterofthefirsteditionwetouchedbrieflyonhowtheonethousand year old calculus problem of Integration in Closed Form which dates back to NewtonandLeibniz,thefoundersofcalculus,wassolvedusingabstractalgebra.In this edition the explanation is significantly expanded by introducing the key con- cept of differential field. That p and e are transcendental numbers was proved in the first edition. In this edition exciting advances in transcendental number theory which have occurred in the130yearssinceLindemannprovedthatpistranscendentalareincluded.Ithank Taboka Prince Chalebgwa for advice on the section on Transcendental Number Theory. As students were very complimentary about the gentle style of presentation in the first edition, every attempt has been made to maintain that in this edition. Anotherfeatureofthiseditionistheinclusionofmuchmorehistoricalinformation. The bibliography has also been greatly expanded. In several places in the book there are Proofs by Contradiction. To highlight when this type of proof is being used the word “suppose” is reserved for the beginning of a proof by contradiction. In other proofs the word “assume” appears rather than “suppose”. The first edition grew out of a second-year subject which Arthur Jones, Ken Pearson, and I had taught for many years at La Trobe University in Melbourne, Australia. This second edition is aimed at similar students. But it should also be mentioned that the knowledge in this book would be beneficial to future (and v vi PrefacetotheSecondEdition current) high school teachers. The level of the material also means that it is accessible to talented high school students. TheauthorsofthefirsteditionwereArthurJones,KenPearson,andme.Arthur and Ken died in 2006 and 2015, respectively. While writing this second edition, I missed them greatly—each had a great passion for teaching and for mathematics. In preparing this edition I was assisted by the advice from the Editors of this Series. I am also grateful to Springer whose staff provided a LaTeX version of the firsteditionasastartingpointforthesecondedition.Iamextremelygratefultothe Springer Mathematics Editor, Dr Loretta Bartolini, who displayed much profes- sionalism and patience. Lastbutnotleast,IthankmywifeElizabethforhercontinuedsupportoverfour decades of marriage. Bundoora/Ballarat, Australia Sidney A. Morris March 2022 Preface to the First Edition The famous problems of squaring the circle, doubling the cube, and trisecting an anglecapturedtheimaginationofbothprofessionalandamateurmathematiciansfor over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods.Itwasonlythedevelopmentofabstractalgebrainthenineteenthcentury which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. In this book we develop enough abstract algebra to prove that these construc- tionsareimpossible.Ourapproachintroducesalltherelevantconceptsaboutfields in a way which is more concrete than usual and which avoids the use of quotient structures (and even of the Euclidean algorithm for finding the greatest common divisor of two polynomials). Having the geometrical questions as a specific goal provides motivation for the introduction of the algebraic concepts and we have found that students respond very favourably. Wehaveusedthistexttoteachsecond-yearstudentsatLaTrobeUniversityover a period of many years, each time refining the material in the light of student performance. The text is pitched at a level suitable for students who have already taken a course in linear algebra, including the ideas of a vector space over a field, linear independence, basis and dimension. The treatment, in such a course, offields and vector spaces as algebraic objects should provide an adequate background for the study of this book. Hence the book is suitable for Junior/Senior courses in North America and second-year courses in Australia. Chapters 1 to 6, which develop the link between geometry and algebra, are the core of this book. These chapters contain a complete solution to the three famous problems,exceptforprovingthatpisatranscendentalnumber(whichisneededto complete the proof of the impossibility of squaring the circle). In Chapter 7 (Chapter 10 in the second edition) we give a self-contained proof that p is tran- scendental. Chapter 8 (Chapter 11 in the second edition) contains material about fields which is closely related to the topics in Chapters 2–4, although it is not required in the proof of the impossibility of the three constructions. The short vii viii PrefacetotheFirstEdition concludingChapter9(Chapter12inthesecondedition)describessomeotherareas ofmathematicsinwhichalgebraic machinery canbeusedtoprove impossibilities. WeexpectthatanycoursebasedonthisbookwillincludeallofChapters1–6and (ideally)atleastpassingreferencetoChapter9(Chapter12inthesecond edition). Wehaveoftentaughtsuchacoursewhichwecoverinaterm(abouttwentyhours). Wefinditessentialforthecoursetobepacedinawaythatallowstimeforstudents todoasubstantialnumberofproblemsforthemselves.Differentsemesterlength(or longer)coursesincluding topicsfromChapters 7and8(Chapters 11 and12inthe second edition) arepossible. The three natural parts ofthese are (1) Sections 7.1 and 7.2 (Sections 10.1 and 10.2 in the second edition) (tran- scendence of e), (2) Sections7.3to7.6(10.3to10.6.14inthesecondedition)(transcendenceofp), (3) Chapter 8 (Chapter 11 in the second edition). These are independent except, of course, that (2) depends on (1). Possible exten- sions to the basic course are to include one, two or all of these. While most treatmentsofthetranscendenceofprequirefamiliaritywiththetheoryoffunctions of a complex variable and complex integrals, ours in Chapter 7 (Chapter 10 in the secondedition)isaccessibletostudentswhohavecompletedtheusualintroductory real calculus course (first-year in Australia and Freshman/Sophomore in North America).HoweverinstructorsshouldnotethattheargumentsinSections7.3to7.6 (Sections 10.3 to 10.6.14 in the second edition) are more difficult and demanding than those in the rest of the book. Problems are given at the end of each section (rather than collected at the end ofthechapter).Someofthesearecomputationalandothersrequirestudentstogive simple proofs. Each chapter contains additional reading suitable for students and instructors. We hope that the text itself will encourage students to do further reading on some of the topics covered. Asinmanybooks,exercisesmarkedwithanasterisk(cid:3)areagoodbitharderthan the others. We believe it is important to identify clearly the end of each proof and we use the symbol (cid:1) for this purpose. We have found that students often lack the mathematical maturity required to writeorunderstandsimpleproofs.Ithelpsifstudentswritedownwheretheproofis heading,whattheyhavetoprove andhowtheymightbeable toprove it.Because thisisnotpartoftheformalproof,weindicatethisexplorationbyseparatingitfrom the proof proper by using a box which looks like Assume we are asked to prove that some result is true for every positive integer n. We might first look at some special cases such as n¼1, n¼2, n¼3. We may see that the result is true in these three cases. Of course this doesnotmeanitistrueforallpositiveintegersn.Whilewhatwehavewritten maynotformpartofaformalproof,itmayneverthelessgiveusahinttohow to proceed to prove the general result. PrefacetotheFirstEdition ix Experience has shown that it helps students to use this material if important theorems are given specific names which suggest their content. We have enclosed these namesinsquare brackets before thestatementofthetheorem. Weencourage studentstousethesenameswhenjustifyingtheirsolutionstoexercises.Theyoften finditconvenienttoabbreviatethenamestojusttherelevantinitials.(Forexample, the name “Small Degree Irreducibility Theorem” can be abbreviated to S.D.I.T.) We are especially grateful to our colleague Gary Davis, who pointed the way towards a more concrete treatment of field extensions (using residue rings rather than quotient rings) and thus made the course accessible to a wider class of stu- dents. We are grateful to Ernie Bowen, Jeff Brooks, Grant Cairns, Mike Canfell, BrianDavey,Alistair Gray,PaulHalmos,PeterHodge,AlwynHoradam,Deborah King, Margaret McIntyre, Bernhard H. Neumann, Kristen Pearson, Suzanne Pearson, Alf van der Poorten, Brailey Sims, Ed Smith and Peter Stacey, who have given us helpful feedback, made suggestions and assisted with the proof reading. We thank Dorothy Berridge, Ernie Bowen, Helen Cook, Margaret McDonald andJudyStoreyforskilfulTeXingofthetextanddiagrams,andNormanGaywood for assisting with the index. April 1991 Arthur Jones Sidney A. Morris Kenneth R. Pearson
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