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The Mathematical Writings of Evariste Galois

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The Mathematical Writings of Evariste Galois 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: +41 (0)44 632 34 36 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 À mes chères amies Jackie Stedall, Sabine Rommevaux, la Ville de Paris, et la Bibliothèque de l’Institut de France A family sketch of Évariste Galois aged 15. First published in 1896 by Paul Dupuy. 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 twelve to thirteen months surrounding his eighteenth birthday. An article published in June 1830 created the theory of Galois imaginaries, a fore-runner of what are now known as finite fields; his so-called Premier Mémoire created group theory and Galois Theory—the modern version of the theory of equations. The Lettre testamentaire, the letter that he wrote to his friend Auguste Chevalier on 29 May 1832, the eve of the duel, is an extraordinary summary of what he had achieved and what he might have achieved had he lived to develop and expound more of his mathematical ideas. Although there have been several French editions of his writings, there has never until now been a systematic English translation. Translations of historical material are of little use without the originals alongside, however. What is offered here therefore is a bilingual edition. The French transcription is a new one. Following precedents set by Tannery in 1906/07 and by Bourgne & Azra in 1962 it is as close to the original manuscripts as I have been able to make it. Main text, afterthoughts, deletions, insertions, over-writings—all are recorded as faithfully as I could manage within the inevitable constraints imposed by the differences between manuscript and print. 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 on the disposition of their content; third, there are comparisons of the various previ- ous editions, including variant readings, in minutely pedantic and minutely printed marginal notes. Little of the commentary here is mathematical. It is focussed on the symbols on the page, on the syntax, on establishing an accurate text. Commen- taries on the semantics, the meaning of what Galois wrote, would be a quite different exercise. That comes next, but must be the subject of other studies. I have neither the space nor the time. Space is a concern because the book is already substantially longer than I had anticipated in light of the shortness of Galois’ productive life. Time is short because a proper modern study of his writings would take years, whereas it is planned that this book should appear on 25 October 2011 as homage to Galois on the 200th anniversary of his birth. The book is conceived as a contribution to the history of mathematics. I hope, however, that it may bring the mathematical writings of this extraordinary genius to 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 and more elegant than what has now become textbook tradition. The details of what 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 constraints it will have succeeded. Acknowledgements It is a pleasure to be able to publish my very warm thanks to a large number of people without whose help and advice this book would have been greatly the poorer. First come Mme Mireille Pastoureau, Director of the Library of the Institut de France, 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 Mme Annie Chassagne, as having earned my special thanks for the many times they have given me special help, 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, Mme Hélène Carrère d’Encausse, Secrétaire perpétuel de l’Académie française, for kind permission to include images of some of the Galois manuscripts in this work; these thanks extend to Mme Florence Greffe, Archivist of the Académie de Sciences, 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 towards the purchase of digital images of the manuscripts and I thank them also. 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, sent pertinent and very helpful criticisms. It is a great pleasure to be able to thank them publicly, not only for their kind comments and help, but also for agreeing to waive the conventional anonymity that usually prevents such thanks and acknowledgment. Jackie Stedall and Sabine Rommevaux also checked the whole work for me—they know already how grateful I am. I am grateful also to Catherine Goldstein and Adrian Rice for bibliographical advice, and to Tessa Shaw and Lynette Dobson of the Queen’s College library for detailed and highly skilled bibliographical assistance. Professor Roger Pearson, FBA, Tutor in French at The Queen’s College, offered help with the mottos and some other mysterious constructions. I am not the first, nor will I be the last, to be grateful for his great scholarship and tutorial skills. Throughout the whole project my editors Manfred Karbe and his wife Irene Zim- mermann have provided an extraordinary level of support and encouragement, freely and patiently sharing with me their technical and TEXnical expertise, experience and wisdom. One could not possibly have a better editorial team. …MN Oxford, September 2011 Contents Preface vii List of facsimiles xi I Introduction 1 I.1 Évariste Galois 1811–1832, revolutionary mathematician .... 1 I.2 What Galois might have read ................... 4 I.3 The manuscripts .......................... 6 I.4 Publication history of Galois’ mathematical writings ....... 8 I.5 The reception of Galois’ ideas ................... 10 I.6 Scope of this edition ........................ 11 I.7 Editorial ambition and policy ................... 12 I.7.1 Auguste Chevalier ...................... 12 I.7.2 Joseph Liouville ....................... 13 I.7.3 Jules Tannery ........................ 14 I.7.4 Robert Bourgne & Jean-Pierre Azra ............ 15 I.7.5 The present work ...................... 16 I.8 Translation and interpretation ................... 17 I.8.1 The words analyste, géomètre ................ 18 I.8.2 The phrases équation algébrique, équation numérique ... 19 I.8.3 The words permutation, substitution ...........
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