Table Of ContentSpringer 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
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Introduction
Theconclusion,ifIamnotmistaken,isthataboveallthe
moderndevelopmentofpuremathematicstakesplaceunderthe
bannerofnumber.
DavidHilbert,TheTheoryofAlgebraicNumberFields,p.ix.
Introduction totheHistoryof 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.
