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Springer Undergraduate Mathematics Series Jeremy Gray A History of Abstract Algebra From Algebraic Equations to Modern Algebra Springer Undergraduate Mathematics Series AdvisoryBoard M.A.J.Chaplain,UniversityofSt.Andrews A.MacIntyre,QueenMaryUniversityofLondon S.Scott,King’sCollegeLondon N.Snashall,UniversityofLeicester E.Süli,UniversityofOxford M.R.Tehranchi,UniversityofCambridge J.F.Toland,UniversityofBath Moreinformationaboutthisseriesathttp://www.springer.com/series/3423 Jeremy Gray A History of Abstract Algebra From Algebraic Equations to Modern Algebra 123 JeremyGray SchoolofMathematicsandStatistics TheOpenUniversity MiltonKeynes,UK MathematicsInstitute UniversityofWarwick Coventry,UK ISSN1615-2085 ISSN2197-4144 (electronic) SpringerUndergraduateMathematicsSeries ISBN978-3-319-94772-3 ISBN978-3-319-94773-0 (eBook) https://doi.org/10.1007/978-3-319-94773-0 LibraryofCongressControlNumber:2018948208 MathematicsSubjectClassification(2010):01A55,01A60,01A50,11-03,12-03,13-03 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Introduction Theconclusion,ifIamnotmistaken,isthataboveallthe moderndevelopmentofpuremathematicstakesplaceunderthe bannerofnumber. DavidHilbert,TheTheoryofAlgebraicNumberFields,p.ix. Introduction to theHistory of Modern Algebra This book covers topics in the history of modern algebra. More precisely, it looks at some topics in algebra and number theory and follows them from their modest presence in mathematics in the seventeenth and eighteenth centuries into the nineteenth century and sees how they were gradually transformed into what we call modernalgebra.Accordingly,it looksatsome of the greatsuccess stories in mathematics: Galois theory—the theory of when polynomial equations have algebraicsolutions—andalgebraicnumbertheory.Soitconfrontsaquestionmany studentsask themselves:howis itthatuniversity-levelalgebrais so verydifferent fromschool-levelalgebra? Theterm‘modernalgebra’wasdecisivelyintroducedbyvanderWaerdeninhis bookModerneAlgebra(1931),anditisworthdiscussingwhathemeantbyit,and whatwas‘modern’aboutit.Thatitstillsufficesasanaccuratelabelformuchofthe workdonein the field since is indicativeof howpowerfulthe movementwas that createdthesubject. Theprimarymeaningoftheterm‘modernalgebra’isstructuralalgebra:thestudy ofgroups,rings,andfields.Interestingly,itdoesnotusuallyincludelinearalgebra orfunctionalanalysis,despitethestronglinksbetweenthesebranches,andapplied mathematiciansmaywellencounteronlyafirstcourseingroupsandnothingelse. Modernalgebraisinmanywaysdifferentfromschoolalgebra,asubjectwhosecore consistsoftheexplicitsolutionofequationsandmodestexcursionsintogeometry. The reasons for the shift in meaning, and the ways in which structural algebra grewoutofold-styleclassicalalgebra,areamongthemajorconcernsofthisbook. Elucidating these reasons, and tracing the implications of the transformation of algebra,will take usfromthelater decadesofthe eighteenthcenturyto the 1920s andthemilieuinwhichModerneAlgebrawaswritten. v vi Introduction Classical algebra confronted many problems in the late eighteenth century. Among them was the so-called fundamental theorem of algebra, the claim that every polynomial with real (or complex) coefficients, has as many roots as its degree. Because most experience with polynomial equations was tied to attempts to solve them this problem overlappedwith attempts to find explicit formulae for theirsolution,andoncethepolynomialhaddegree5ormorenosuchformulawas known. The elusive formula was required to involve nothing more than addition, subtraction,multiplication,anddivisionappliedto the coefficientsof the equation andtheextractionofnthroots,soitwasknownassolutionbyradicals,andLagrange in1770wasthefirsttogivereasonswhythegeneralquinticequationmightnotbe solvablebyradicals. Anothersourceofproblemsinclassicalalgebrawasnumbertheory.Fermathad triedandlargelyfailedtointeresthiscontemporariesinthesubject,buthiswritings on the subjectcaughtthe attentionof Euler in the eighteenthcentury.Euler wrote extensivelyonthem,andwhatheconjecturedbutcouldnotprovewasoften,butnot always,soonprovedbyLagrange.Thisleftarangeofpartiallyansweredquestions for their successors to pursue. For example, Fermat had shown that odd prime numbersoftheformx2+y2arepreciselythoseoftheform4k+1,andhadfound similartheoremsforprimesoftheformx2+2y2 andx2+3y2 butnotforprimes of the formx2 +5y2—whynot, whatwas goingon? Therewas a goodtheoryof integersolutionstotheequationx2−Ay2 =1,whereAisasquare-freeinteger,and Lagrangehadbegunatheoryofthegeneralbinaryquadraticform,ax2+bxy+cy2, butmuchremainedtobedone.And,famouslytodayiflesssoin1800,Fermathad aconjectureaboutintegersolutionstoxn+yn = zn,n > 2andhadindeedshown that there were no solutions when n = 4, and Euler had a suggestive but flawed proofofthecasex3+y3 =z3thatledintothetheoryofquadraticforms. Thenineteenthcenturybegan,inalgebra,withGauss’sDisquisitionesArithmeti- cae (1801), the book that made Gauss’s name and may be said to have created modernalgebraicnumbertheory,inthesensethatitinspiredanunbrokenstreamof leadingGermanmathematiciansto take upand developthe subject.1 Theworkof GaussandlaterDedekindiscentraltothestoryofthecreationofmodernalgebra. In the 1820s Abel had wrapped up the question of the quintic, and shown it was not generally solvable by radicals. This raised a deeper question, one that Gauss hadalreadybeguntoconsider:giventhatsomepolynomialequationsofdegree5or morearesolvablebyradicals,whichonesareandhowcanwetell?Itwasthegreat, ifobscure,achievementofGaloistoshowhowthisquestioncanbeanswered,and theimplicationsofhisideas,manyofthemdrawnoutbyJordanin1870,alsouncoil throughthenineteenthcenturyandfigurelargelyinthestory. Algebraic number theory led mathematicians to the concept of commutative rings,ofwhich,afterall,theintegersarethecanonicalexample,andGaloistheory ledtotheconceptsofgroupsandfields.Otherdevelopmentsinnineteenthcentury geometry—the rediscovery of projective geometry and the shocking discovery of 1Gaussalsorediscoveredthefirstknownasteroidin1801. Introduction vii non-Euclidean geometry—also contributed to the success of the group concept, whenKleinuseditinthe1870stounifythedisparatebranchesofgeometry. The rise of structural mathematics was not without controversy. There was a long-runningargumentbetween Kroneckerand Dedekind aboutthe propernature of algebraic number theory. There was less disagreement about the importance of Galois’s ideas once they were properly published, 14 years after his untimely death,butittookagenerationtofindtherightwaytohandlethemandanotherfor the modern consensus to emerge. By the end of the century there was a marked disagreementinthemathematicalcommunityabouttherelativeimportanceofgood questionsandabstracttheory.Thisisnotjustachicken-and-eggproblem.Onceitis agreedthattheoryhasamajorplace,itfollowsthatpeoplecanworkontheoryalone, andthesubjecthastogrowtoallowthat.Byandlarge,equationshavecontextsand solvingthemisofvalueinthatcontext,butwhatisthepointofatheoryofgroups whendoneforitsownsake?Questionsaboutintegersmaybeinteresting,butwhat aboutan abstracttheoryof rings? These questionsacquiredsolid answers, onesit is the historians’ job to spell out, but they are legitimate, as are their descendants today:highercategorytheory,anyone? The end of the nineteenth century and the start of the 20th see the shift from classical algebra to modern algebra, in the important sense that the structural conceptsmovefromtheresearchfrontiertothecoreandbecomenotonlythewayin whichclassical problemscanbe reformulatedbuta sourceof legitimate problems themselves. This was a lengthy process, and the publication of Moderne Algebra marks an important stage in what is still an ongoing process, one that is worth thinkingabout.LeoCorry(1996)hasusefullydistinguishedbetweenwhathecalls the ‘body’ and the ‘images’ of a piece of knowledge: the body of knowledge, he tells us (1996, 3) “includes theories, ‘facts’, open problems”, the images “serve as guiding principles, or selectors”, they “determine attitudes” about what is an urgentproblemdemandingattention,whatisrelevant,whatisalegitimatemethod, what should be taught, and who has the authority to decide. It might indeed be worthseparatingoutthesocialandinstitutionalfactorsentirelythatcouldbecalled the ‘forces of knowledge’: the mathematicians themselves and their institutions (universities,professionalbodies,journals,andthelike). I have tried to introduce students to some of the best writing in the history of mathematics of the last 20 or 30 years, not just for the information these books andpapersprovidebutasintroductionstohowhistorycanbewritten,andto help students engage with other opinions and so form their own. The most relevant examples are the The Shaping of Arithmetic after C.F. Gauss’s Disquisitiones Arithmeticae by Goldstein et al. (2007) and Corry’s Modern algebra and the rise of mathematical structures (1996), both of which in their different ways show how much can be done when one gets away from the monotonous plod of great achievements.Kiernan’sinfluentialpaper‘ThedevelopmentofGaloistheoryfrom Lagrange to Artin’ (Kiernan 1971) is still well worth consulting. Such accounts viii Introduction are analytical, not merely descriptive. As Gauss famously said of his own work2 “Whenthebuildingisfinishedthescaffoldingcannolongerbeseen”;whateverthe meritsanddemeritsofpresentingmathematicsthatwaylittle isgainedbytreating thehistoryofmathematicsasanattempttorebuildeachbranchofmathematicsas itistoday,brickbybrick,withnamesanddatesattached. Inshort,thisisabookonthehistoryofalgebra,andassuchitasksthestudent tothinkaboutwhatitis tostudyhistory.Itemphasisessome pointsthata straight mathematics course might marginalise, and it omits others that a mathematician would emphasise, as and when I judged that a history course required it. It does assumethatthestudentscanhandledifficultmathematics—andhappilythatproved generallytobethecaseintheyearsItaughtitattheUniversityofWarwick—butit requiresthattheymarshalarguments,basedonfacts,andinsupportofopinions,as historiansmust.Ihavetriedtomakeitcoherent,albeitselective. It is not an attempt to write a ‘complete’ history of algebra in the nineteenth century.Itistheresultofacourseof30lectures,anditcanbetaughtassuch.Not allthechapterscorrespondexactlytoalecture;someovershoot,andthereismore information in the Appendices. To reduce the study of algebra in the nineteenth centuryandtheearlyyearsofthe20thtobarely300pagesmeanttakingsomecrude decisions,and I couldsee no tidy way to do it. Algebrain the period,it seems, is a more heterogeneous body of knowledge than geometry was, and even real and complexanalysis(oncedifferentialequationsofallkindsarereservedforanother occasion). I removedmost of invarianttheory and the Kleinian view of geometry from discussion. The history of work on determinants, matrices, linear algebra generally,quaternionsandotheralgebrashadalsotobeomitted.Butthatonlyleft me facing tougher decisions. I decided not to deal with Galois’s second memoir, Sophie Germain and her work, higher reciprocity laws, Fermat’s Last Theorem, powerseriesmethods,andthedistributionofprimenumbers.Severalofthesetopics arewellcoveredintheexistingliteratureandwouldmakegoodprojectsforstudents wishing to take the subject further. Although it has become increasingly clear in recentyearsthatthe workofKummer,Kronecker,Hermiteand othersontherich andoverlappingfieldsofellipticfunctions,modularfunctionsandn-aryquadratic formswasamajorlegacyofGauss,noneofthatcouldbedescribedhere.Infactit makesasuitablesubjectforresearch.3Icouldnot,however,resistwritingaboutthe importanceofKlein’sfamousbutseldom-readLecturesontheIcosahedron(1884). No undergraduate history course assumes that the students will be competent, let alone masterful, in everything that matters to the course. Students study the techniques of warfare without being able to ride a horse or fire a gun; diplomacy without being taught the skills of negotiation; nationalism without engaging in politics;labourmovementswithoutworkingin a field or a factory;childhoodand 2SeeSartorius(1856,82). 3On Hermite, see Goldstein (2007) and Goldstein (2011); on Kronecker see Goldstein and Schappacher (2007, Section4)andontheKronecker–Kleindisputesee(PetriandSchappacher 2002). Introduction ix thefamilywithoutbecomingpregnantorrearingchildren.Butthereisafeelingthat the history of a branch of mathematics should not be taught without the students acquiringsomethinglikethemasteryofthatbranchthatastraightcourseinitwould hope to achieve. This book is an attempt not to break that connection—students shouldnotsaynonsenseaboutquadraticreciprocityorwhichequationsaresolvable byradicals—buttoopenituptootherapproaches. Theethicsherearethoseofanappliedmathematicscourse,inwhichabalance istobestruckbetweenthemathematicsandtheapplication.Doubtlessthisbalance isstruckinmanydifferentwaysindifferentcourses.HereIhavetriedtobringout whatisimportanthistoricallyinthedevelopmentofthisorthatpartofalgebra,and to do so it is necessary to take some things for granted, say, that this person did validly deduce this result although the proof will not be looked at. The historical purposeinexaminingastrictlymathematicaldetailhastobedecidedcasebycase. Togiveoneexample,perhapsthegravestinthebook,thetreatmentofquadratic reciprocity given here contains one complete proof of the general case (Gauss’s thirdprooffrom1808in whichhe introducedGauss’s Lemma)andindicateshow he gave an earlier proof, in the Disquisitiones Arithmeticae, that rested on his techniqueof compositionof forms(althoughI discuss the fourth and sixth proofs in AppendixC). To explaincompositionof formsI indicate whatis involved,and give Dirichlet’s simplified account, but I merely hint at the difficulties involved in securing the deepest results without which Gauss’s theory of quadratic form s losesmuchofitsforce.Thisgivesmeroomtoexplainthehistoryofthisfamously difficultsubject,butdeprivesthestudentsofsomekeyproofs.My judgementwas thatthiswasenoughforahistoryofmodernalgebrainwhichthecontemporaryand subsequentappreciationoftheseideasisatleastasimportantastheaccompanying technicalities.IthengoontodescribeDedekind’stranslationofthetheoryintothe theoryofmodulesandidealsthathecreated. I alsobelievethatsufficientunderstandingofsometopicsforthepurposesofa historycourseisacquiredbyworkingwithexamples,andwhenthesearenotmade explicitthereisalwaysanimplicationthatexampleshelp. In1817Gausswrote(seeWerke2,pp.159–160)that It is characteristic of higher arithmetic that many of its most beautiful theorems can be discoveredbyinductionwiththegreatestofeasebuthaveproofsthatlieanywherebutnear at handandareoften found onlyafter manyfruitlessinvestigationswiththeaidofdeep analysisandluckycombinations Certainly numerical exploration and verification are a good way to understand manyoftheideasinthisbook.Thattheproofsliedeepbelowthesurfaceisoneof theprincipalimpulsesfortheslowelaborationofmodernalgebra.