The mathematical writings of Evariste Galois

422 Pages·2011·3.285 MB·English

by  Peter M. Neumann

Heritage of European Mathematics
Advisory Board
Ciro Ciliberto, Roma
Ildar A. Ibragimov, St. Petersburg
Wladyslaw Narkiewicz, Wroclaw
Peter M. Neumann, Oxford
Samuel J. Patterson, Göttingen
Previously published
Andrzej Schinzel, Selecta,
Volume I: Diophantine Problems and Polynomials;
Volume II: Elementary, Analytic and Geometric Number Theory,
Henryk Iwaniec, Wladyslaw Narkiewicz, and Jerzy Urbanowicz(Eds.)
Thomas Harriot’s Doctrine of Triangular Numbers: the ‘Magisteria Magna’,
Janet Beery and Jacqueline Stedall (Eds.)
Hans Freudenthal, Selecta,Tony A. Springer and Dirk van Dalen (Eds.)
Nikolai I. Lobachevsky, Pangeometry, Athanase Papadopoulos (Transl. and Ed.)
Jacqueline Stedall, From Cardano’s great art to Lagrange’s reflections: filling a
gap in the history of algebra
Peter M. Neumann
The mathematical
writings of
Évariste Galois
Author:
Peter M.Neumann
The Queen’s College
Oxford,OX1 4AW
United Kingdom
E-mail:[email protected]
2010 Mathematics Subject Classification (primary;secondary):01-02;01-55,01A75,00B55,11-03,
11A55,12-03,12E12,12E20,12F10,20-02,20-03,20B05,20B15,20D05,33-03,33E05
Key words:History of mathematics, Galois,Galois Theory,group,Galois group,equation,theory of
equations,Galois field,finite field,elliptic function,modular equation,primitive equation,primitive
group,solubility,simple group,soluble group
ISBN 978-3-03719-104-0
The Swiss National Library lists this publication in The Swiss Book,the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright.All rights are reserved,whether the whole or part of the material is
concerned,specifically the rights of translation,reprinting,re-use of illustrations,recitation,broad-
casting,reproduction on microfilms or in other ways,and storage in data banks.For any kind of use
permission of the copyright owner must be obtained.
© 2011 European Mathematical Society
Contact address:
European Mathematical Society Publishing House
Seminar for Applied Mathematics
ETH-Zentrum FLI C4
CH-8092 Zürich
Switzerland
Phone:[phone]
Email:[email protected]
Homepage:www.ems-ph.org
Typeset using the author’s TEX files:I.Zimmermann,Freiburg
Printing and binding:Beltz Bad Langensalza GmbH,Bad Langensalza,Germany
∞Printed on acid free paper
9 8 7 6 5 4 3 2 1
Àmeschèresamies
JackieStedall,
SabineRommevaux,
laVilledeParis,
et
laBibliothèquedel’InstitutdeFrance
AfamilysketchofÉvaristeGaloisaged15.
Firstpublishedin1896byPaulDupuy.
Preface
Before he died aged twenty, shot in a mysterious early-morning duel at the end
of May 1832, Évariste Galois created mathematics which changed the direction of
algebra. His revolutionary ideas date from around May 1829 to June 1830, the
twelvetothirteenmonthssurroundinghiseighteenthbirthday. Anarticlepublished
inJune1830createdthetheoryofGaloisimaginaries,afore-runnerofwhatarenow
knownasfinitefields;hisso-calledPremierMémoirecreatedgrouptheoryandGalois
Theory—the modern version of the theory of equations. The Lettre testamentaire,
the letter that he wrote to his friendAuguste Chevalier on 29 May 1832, the eve of
the duel, is an extraordinary summary of what he had achieved and what he might
haveachievedhadhelivedtodevelopandexpoundmoreofhismathematicalideas.
AlthoughtherehavebeenseveralFrencheditionsofhiswritings,therehasnever
untilnowbeenasystematicEnglishtranslation. Translationsofhistoricalmaterialare
oflittleusewithouttheoriginalsalongside,however. Whatisofferedherethereforeis
abilingualedition. TheFrenchtranscriptionisanewone. Followingprecedentsset
byTanneryin1906/07andbyBourgne&Azrain1962itisasclosetotheoriginal
manuscripts as I have been able to make it. Main text, afterthoughts, deletions,
insertions,over-writings—allarerecordedasfaithfullyasIcouldmanagewithinthe
inevitableconstraintsimposedbythedifferencesbetweenmanuscriptandprint.
In addition I offer three levels of commentary. First there is general contextual
information; secondly there are notes on the physical state of the manuscripts and
onthedispositionoftheircontent;third,therearecomparisonsofthevariousprevi-
ous editions, including variant readings, in minutely pedantic and minutely printed
marginal notes. Little of the commentary here is mathematical. It is focussed on
thesymbolsonthepage,onthesyntax,onestablishinganaccuratetext. Commen-
tariesonthesemantics,themeaningofwhatGaloiswrote,wouldbeaquitedifferent
exercise. That comes next, but must be the subject of other studies. I have neither
thespacenorthetime. Spaceisaconcernbecausethebookisalreadysubstantially
longerthanIhadanticipatedinlightoftheshortnessofGalois’productivelife. Time
isshortbecauseapropermodernstudyofhiswritingswouldtakeyears,whereasit
isplannedthatthisbookshouldappearon25October2011ashomagetoGaloison
the200th anniversaryofhisbirth.
The book is conceived as a contribution to the history of mathematics. I hope,
however,thatitmaybringthemathematicalwritingsofthisextraordinarygeniusto
a wider mathematical public than has hitherto been able to appreciate them. At the
very least it may serve to dispel some of the common myths that surround Galois
and his understanding of mathematics. It is simply not true, for example, that he
proved and used the simplicity of alternating groups. He did not need to: he was
much cleverer than that; his treatment of solubility of equations is at once simpler
andmoreelegantthanwhathasnowbecometextbooktradition. Thedetailsofwhat
he did, the proper evidence of his genius, deserve to be as well understood and
appreciated amongst mathematicians as amongst historians of mathematics. If this
viii Preface
edition extends his readership beyond the bounds presently imposed by linguistic
constraintsitwillhavesucceeded.
Acknowledgements
Itisapleasuretobeabletopublishmyverywarmthankstoalargenumberofpeople
without whose help and advice this book would have been greatly the poorer. First
comeMmeMireillePastoureau,DirectoroftheLibraryoftheInstitutdeFrance,and
her staff. They have all accorded me the highest level of kindness and assistance.
If I pick out two people, Mme Fabienne Queyroux and MmeAnnie Chassagne, as
havingearnedmyspecialthanksforthemanytimestheyhavegivenmespecialhelp,
I hope it will not detract from my thanks to all their colleagues. I thank also the
Commission des bibliothèques et archives de l’Institut de France, and its president,
MmeHélèneCarrèred’Encausse,Secrétaireperpétueldel’Académiefrançaise,for
kind permission to include images of some of the Galois manuscripts in this work;
thesethanksextendtoMmeFlorenceGreffe,ArchivistoftheAcadémiedeSciences,
who has made available an image of Galois’letter of 31 March 1831 and a pam-
phlet by Jacques Tits. Professor Jean-Pierre Kahane, a member of that committee,
has been very supportive and it is a pleasure to record my personal thanks to him.
Mr F. Xavier Labrador of the Société d’Ingénierie et de Microfilmage made those
high quality photographs, and I am very grateful to him. I thank also Jonathan
Crayford for photographs and much enthusiasm about Galois and his work. The
Governing Body of the Queen’s College, Oxford, and the Research Committee of
the Mathematical Institute in the University of Oxford provided financial support
towardsthepurchaseofdigitalimagesofthemanuscriptsandIthankthemalso.
Several colleagues read an early draft of the book, or parts of it, and I have
benefitted from their suggestions. Two referees, Massimo Galuzzi and Caroline
Ehrhardt,sentpertinentandveryhelpfulcriticisms. Itisagreatpleasuretobeable
to thank them publicly, not only for their kind comments and help, but also for
agreeingtowaivetheconventionalanonymitythatusuallypreventssuchthanksand
acknowledgment. Jackie Stedall and Sabine Rommevaux also checked the whole
workforme—theyknowalreadyhowgratefulIam. IamgratefulalsotoCatherine
GoldsteinandAdrianRiceforbibliographicaladvice,andtoTessaShawandLynette
DobsonoftheQueen’sCollegelibraryfordetailedandhighlyskilledbibliographical
assistance. ProfessorRogerPearson,FBA,TutorinFrenchatTheQueen’sCollege,
offeredhelpwiththemottosandsomeothermysteriousconstructions. Iamnotthe
first,norwillIbethelast,tobegratefulforhisgreatscholarshipandtutorialskills.
ThroughoutthewholeprojectmyeditorsManfredKarbeandhiswifeIreneZim-
mermannhaveprovidedanextraordinarylevelofsupportandencouragement,freely
andpatientlysharingwithmetheirtechnicalandTEXnicalexpertise,experienceand
wisdom. Onecouldnotpossiblyhaveabettereditorialteam.
…MN Oxford,September2011
Contents
Preface vii
Listoffacsimiles xi
I Introduction 1
I.1 ÉvaristeGalois1811–1832,revolutionarymathematician . . . . 1
I.2 WhatGaloismighthaveread . . . . . . . . . . . . . . . . . . . 4
I.3 Themanuscripts . . . . . . . . . . . . . . . . . . . . . . . . . . 6
I.4 PublicationhistoryofGalois’mathematicalwritings . . . . . . . 8
I.5 ThereceptionofGalois’ideas . . . . . . . . . . . . . . . . . . . 10
I.6 Scopeofthisedition . . . . . . . . . . . . . . . . . . . . . . . . 11
I.7 Editorialambitionandpolicy . . . . . . . . . . . . . . . . . . . 12
I.7.1 AugusteChevalier . . . . . . . . . . . . . . . . . . . . . . 12
I.7.2 JosephLiouville . . . . . . . . . . . . . . . . . . . . . . . 13
I.7.3 JulesTannery . . . . . . . . . . . . . . . . . . . . . . . . 14
I.7.4 RobertBourgne&Jean-PierreAzra . . . . . . . . . . . . 15
I.7.5 Thepresentwork . . . . . . . . . . . . . . . . . . . . . . 16
I.8 Translationandinterpretation . . . . . . . . . . . . . . . . . . . 17
I.8.1 Thewordsanalyste,géomètre . . . . . . . . . . . . . . . . 18
I.8.2 Thephraseséquationalgébrique,équationnumérique . . . 19
I.8.3 Thewordspermutation,substitution . . . . . . . . . . . . 20
I.8.4 Thewordgroupe . . . . . . . . . . . . . . . . . . . . . . 22
I.8.5 Thewordsemblable . . . . . . . . . . . . . . . . . . . . . 23
I.8.6 Thewordprimitif . . . . . . . . . . . . . . . . . . . . . . 24
I.8.7 Otherwordsandphrases . . . . . . . . . . . . . . . . . . 25
I.8.8 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
I.9 A‘wartsandall’transcription . . . . . . . . . . . . . . . . . . . 27
I.10 Thetranscription: editorialconventions . . . . . . . . . . . . . . 31
II Thepublishedarticles 33
II.1 Atheoremoncontinuedfractions . . . . . . . . . . . . . . . . . 35
II.2 Abstractofanarticleonsolutionofequations . . . . . . . . . . 49
II.3 Anoteonthenumericalsolutionofequations . . . . . . . . . . 55
II.4 Onthetheoryofnumbers . . . . . . . . . . . . . . . . . . . . . 61
II.5 Onsomepointsofanalysis . . . . . . . . . . . . . . . . . . . . 77
III TheTestamentaryLetterof29May1832 83
III.1 Theletter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
III.2 Notesontheletter . . . . . . . . . . . . . . . . . . . . . . . . . 101
x Contents
IV TheFirstMemoir 105
IV.1 TextoftheFirstMemoir . . . . . . . . . . . . . . . . . . . . . . 105
IV.2 NotesontheFirstMemoir . . . . . . . . . . . . . . . . . . . . . 145
IV.3 Letterof31March1831totheAcademy . . . . . . . . . . . . . 165
V TheSecondMemoir 169
V.1 TextoftheSecondMemoir . . . . . . . . . . . . . . . . . . . . 169
V.2 NotesontheSecondMemoir . . . . . . . . . . . . . . . . . . . 195
VI Theminormathematicalmanuscripts 199
VI.1 Dossier6: An1830versionofPropositionI . . . . . . . . . . . 201
VI.2 Dossier7: An1830draftofPropositionV . . . . . . . . . . . . 209
VI.3 Dossier8: Atornfragment . . . . . . . . . . . . . . . . . . . . 219
VI.4 Dossier9: Preliminarydiscussion . . . . . . . . . . . . . . . . . 225
VI.5 Dossier10: PublicationprojectandnoteonAbel . . . . . . . . . 233
VI.6 Dossier11: Prefacefortwomemoirs . . . . . . . . . . . . . . . 245
VI.7 Dossier12: Ontheprogressofpureanalysis . . . . . . . . . . . 259
VI.8 Dossier13: Here,asinallthesciences . . . . . . . . . . . . . . 269
VI.9 Dossier14: Science,Hierarchy,Schools . . . . . . . . . . . . . 275
VI.10 Dossier15: Fragmentsonpermutationsandequations . . . . . . 279
VI.11 Dossier16: FragmentsrelatingtoPropositionI . . . . . . . . . . 301
VI.12 Dossier17: Fragmentsonthetheoryofequations . . . . . . . . 311
VI.13 Dossier18: Noteonnon-primitiveequations . . . . . . . . . . . 323
VI.14 Dossier19: Additiontothememoironequations . . . . . . . . . 329
VI.15 Dossier20: Onthedivisionofellipticfunctions . . . . . . . . . 335
VI.16 Dossier21: Ontheintegrationoflinearequations . . . . . . . . 347
VI.17 Dossier22: Onsurfacesoftheseconddegree . . . . . . . . . . . 355
VI.18 Dossier23: Oneulerianintegrals . . . . . . . . . . . . . . . . . 367
VI.19 Dossier24: AtheoremofAbel . . . . . . . . . . . . . . . . . . 373
VII Epilogue: mythsandmysteries 383
VII.1 Myths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
VII.2 Mysteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
VII.3 Lastwords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
Bibliography 391
Index 405