DOCSLIB.ORG

B.L. Van Der Waerden a History of Algebra B.L

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
B.L. Van Der Waerden a History of Algebra B.L B.L. van der Waerden A History of Algebra B.L. van der Waerden A History of Algebra From al-Khwärizml to Emmy Noether With 28 Figures Springer-Verlag Berlin Heidelberg GmbH Prof. Dr. Bartel Leenert van der Waerden Mathematisches Institut der Universität Zürich Mathematics Subject Classification (1980): Ol-XX, lO-XX, l2-XX, l4-XX, l6-XX, l7-XX, 20-XX, 22-XX Library of Congress Cataloging in Publication Data Waerden, B.L. van der (Bartel Leenert), 1903- A history of algebra. Bibliography: p. Includes index. I. Algebra - History. I. Title. QAI5l.W343 1985 512'.009 85-2820 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertunl\sl\esellschaft Wort", Munieh. ISBN 978-3-642-51601-6 ISBN 978-3-642-51599-6 (eBook) DOI 10.1007/978-3-642-51599-6 © by Springer-Verlag Berlin Heidelberg 1985 Originally published by Springer-Verlag Berlin Heidelberg in 1985. 2141/3140-543210 Preface The present volume is self-contained, but it is apart of a larger projecL In an earlier volume, entitled "Geometry and Algebra in Ancient Civilizations", I have presented my view on the prehistory of algebra and geometry from the neolithical period to Brahmagupta (7th century A.D.). The present volume deals with the historical period from al-KhwarizmI, the earliest author of an "Algebra", to Emmy Noether. In this book I shall restrict myself to three subjects, namely Part One: Algebraic Equations; Part Two: Groups; Part Three: Algebras. My warrnest thanks go out to all those who helped me by reading parts of the manuscript and suggesting essential improvements: Wyllis Bandler (Tallahassee, Florida), Robert Burn (Cambridge, England), Max Deuring t (Göttingen), Jean Dieudonne (Paris), Raffaella Franci (Siena), Hans FreudenthaI (Utrecht), Thomas Hawkins (Boston), Erwin Neuenschwander (Zürich), Laura Toti Rigatelli (Siena), Warren Van Egmond (München). In her usual careful way, Miss Annemarie Fellmann has typed the manu­ script. She has also drawn the figures and helped me, together with Erwin Neuenschwander, in reading the proof sheets. Many thanks to both and to the editorial staff and the production department of the Springer-Verlag for their nice cooperation. Zürich, March 1985 B.L. van der Waerden Table of Contents Part One. Algebraic Equations Chapter 1. Three Muslimic Authors 3 Part A. AI-Khwärizmi . 3 1. The Man and his W ork . 3 2. Al-jabr and al-muqabala 4 3. On Mensuration . 5 4. On the Jewish Calendar 7 5. On Legacies . 7 6. The Solution of Quadratic Equations 7 7. The Geography ... 9 8. On Hindu Numerals . 9 9. The Astronomical Tables 9 10. The "Sindhind" . 10 11. The "Method of the Persians" 11 12. AI-Khwärizmi's Sources 13 Part B. Tabit ben Qurra 15 The Sabians ..... 15 The Life of Tabit ben Qurra 16 On the Motion of the Eighth Sphere 17 Geometrical Verification of the Solution of Quadratic Equations 18 On Amicable Numbers . 21 Part C. Omar Khayyam 24 The Algebra of Omar Khayyam 24 Omar Khayyam on Ratios 29 Chapter 2. Algebra in Italy 32 Part A. From Leonardo da Pisa to Luca Pacioli 32 The Connection Between Trade and Civilization in Medieval Italy 32 Life and Work of Fibonacci 33 1. The "Liber Abbaci" . 35 2. The" Practica geometriae" 39 3. The Book "Flos" . 40 4. The Letter to Theodorus . 40 5. The "Liber quadratorum" 40 VIII Table of Contents Three Florentine Abbaeists 42 1. Maestro Benedetto 42 2. Maestro Biaggio 43 3. Antonio Mazzinghi 44 Two Anonymous Manuscripts 45 Luca Pacioli ...... 46 Part B. Master Dardi of Pisa 47 Part C. The Solution of Cubic and Biquadratic Equations 52 Scipione deI Ferro . 52 Tartaglia and Cardano 54 Lodovico Ferrari 56 Rafael Bombelli . 59 Chapter 3. From Viete to Descartes 63 Fran~ois Viete 63 Simon Stevin . 68 Pierre de Fermat 69 Rene Descartes 72 Chapter 4. The Predecessors of Galois 76 Waring 76 Vandermonde 77 Lagrange 79 Malfatti 81 Ruffini 83 Cauchy 85 Abel . 85 Chapter 5. Carl Friedrich Gauss 89 The Cydotomic Equation 89 The "Fundamental Theorem" 94 The First Proof . 95 The Second Proof . 97 The Third Proof 99 Chapter 6. Evariste Galois 103 The Work of Galois 103 The Duel ..... 104 The Memoir of 1831 105 Galois Fields . 109 The Publication of Galois' Papers 112 Hermite, Puiseux, and Serret 112 Enrico Betti ........ 114 The Second Posthumous Memoir of Galois 115 Chapter 7. Camille Jordan 117 Jordan's Traite 117 On Groups of Motions 118 Table of Contents IX On Congruences .......... 121 Transitive and Primitive Groups of Substitutions 121 Series of Composition . 121 Linear Substitutions . 122 Jordan's Presentation of Galois Theory 124 Geometrical Applications ..... 125 The 28 Double Tangents of a Plane Quartic 128 Application of Galois Theory to Transcendental Functions 131 On Solvable Groups . 133 Part Two. Groups Chapter 8. Early Group Theory 137 Part A. Groups of Substitutions . 137 Early Theorems Concerning Subgroups of Sn 137 Mathieu ........... 139 Sylow ............ 139 Part B. Groups of Transformations 140 Non-Euclidean Geometry .... 141 Felix Klein and Sophus Lie . 144 Felix Klein on Finite Groups of Fractional Linear Transformations 145 Sophus Lie ..... 146 Part C. Abstract Groups 147 Leonhard Euler . 147 Carl Friedrich Gauss 148 Ernst Schering 149 Leopold Kronecker 149 Arthur Cayley 150 Walter von Dyck 152 Heinrich Weber . 153 Part D. The Structure of Finite Groups 154 OUo Hölder .......... 155 Finite Linear Groups 158 Chapter 9. Lie Groups and Lie Algebras 160 Part A. Lie Groups . 160 Lie's Theory . 160 Infinitesimal Transformations 162 Three Fundamental Theorems 163 Part B. Lie Aigebras 165 Sophus Lie and Friedrich Engel 166 Wilhelm Killing . 166 Elie Cartan . 168 The Characteristic Roots 168 X Table of Contents Semi-Simple Lie Groups 170 Weyl's Group (S) 172 Real Simple Lie Algebras 173 Part Three. Algebras Chapter 10. The Discovery of Algebras 177 Complex Numbers ...... 177 Hamilton's Discovery of Quaternions 179 The Leap into the Fourth Dimension 181 Octonions .......... 183 Product Formulae for Sums of Squares 184 Geometrical Applications of Quaternions 185 The Arithmetic of Quaternions 186 Biquaternions . 188 Full Matrix Algebras ... 189 Group Algebras .. ... 190 Grassmann's Ca1culus of Extensions 191 Clifford Aigebras and Rotations in n Dimensions 192 Dirac's Theory of the Spinning Electron 194 Spinors in n Dimensions 195 Chevalley's Generalization 197 Generalized Quaternions 197 Crossed Products 199 Cyc1ic Aigebras . 200 Chapter 11. The Structure of Algebras 202 General Notions and Notations 202 Benjamin Peirce . 203 Eduard Study . 204 Gauss, Weierstrass and Dedekind 204 Georg Scheffers 205 Theodor Molien . 206 Elie Cartan . 208 Mac1agan Wedderburn 210 Emil Artin . 210 Emmy Noether and her School 211 Nathan Jacobson . 211 Normal Simple Aigebras . 212 The Structure of Division Aigebras 213 1. Division Algebras over IR. 214 2. Finite Skew Fields 214 3. Normal Simple Aigebras over a P-adic Field 214 4. Division Algebras over an Algebraic Number Field 215 Table of Contents XI Chapter 12. Group Characters 218 Part A. Characters of Abelian Groups . 218 Genera and Characters of Quadratic Forms 218 Duality in Abelian Groups . 221 Part B. Characters of Finite Groups . 223 Dedekind's Introduction of the Group Determinant 223 Frobenius on the Group Determinant 225 Frobenius on Commuting Matrices 229 The Letter of April 17, 1896 .... 231 The Letter of April 26 . 233 Frobenius' Paper "Über Gruppencharaktere" 234 The Proof of e= fand the Factorization of the Group Determinant 236 Chapter 13. Representations of Finite Groups and Algebras 237 Heinrich Maschke . 238 Issai Schur . 239 Representations of the Symmetrie Group . 240 The Representation of Groups by Projective Transformations 243 Emmy Noether . 244 1. Group-Theoretical F oundations 244 2. Non-Commutative Ideal Theory 245 3. Modules and Representations 246 4. Representations of Groups and Algebras 248 Chapter 14. Representations of Lie Groups and Lie Algebras 252 Cartan's Theory . 252 The Global Method . 253 The Infinitesimal Method 254 Hermann Weyl 257 lohn von Neumann 261 Index . 265 .
Load more

Recommended publications
  • A Survey of the Development of Geometry up to 1870
    A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics.
    [Show full text]
  • Symmetry and Mathematics
    A Conceptual History of Space and Symmetry Pietro Giuseppe Fré A Conceptual History of Space and Symmetry From Plato to the Superworld 123 Pietro Giuseppe Fré University of Torino Turin, Italy ISBN 978-3-319-98022-5 ISBN 978-3-319-98023-2 (eBook) https://doi.org/10.1007/978-3-319-98023-2 Library of Congress Control Number: 2018950942 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland This book is dedicated to my family, namely to my beloved daughter Laura, to my darling wife Olga, to my young son Vladimir and to my former wife Tiziana, Laura’s mother, with whom, notwithstanding our divorce, the ties due to a common daughter and to more than twenty five years spent together remain strong.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach the History Of
    Mathematisches Forschungsinstitut Oberwolfach Report No. 51/2004 The History of Differential Equations, 1670–1950 Organised by Thomas Archibald (Wolfville) Craig Fraser (Toronto) Ivor Grattan-Guinness (Middlesex) October 31st – November 6th, 2004 Mathematics Subject Classification (2000): 01A45, 01A50, 01A55, 01A60, 34-03, 35-03. Introduction by the Organisers Differential equations have been a major branch of pure and applied mathematics since their inauguration in the mid 17th century. While their history has been well studied, it remains a vital field of on-going investigation, with the emergence of new connections with other parts of mathematics, fertile interplay with applied subjects, interesting reformulation of basic problems and theory in various periods, new vistas in the 20th century, and so on. In this meeting we considered some of the principal parts of this story, from the launch with Newton and Leibniz up to around 1950. ‘Differential equations’ began with Leibniz, the Bernoulli brothers and others from the 1680s, not long after Newton’s ‘fluxional equations’ in the 1670s. Appli- cations were made largely to geometry and mechanics; isoperimetrical problems were exercises in optimisation. Most 18th-century developments consolidated the Leibnizian tradition, extend- ing its multi-variate form, thus leading to partial differential equations. General- isation of isoperimetrical problems led to the calculus of variations. New figures appeared, especially Euler, Daniel Bernoulli, Lagrange and Laplace. Development of the general theory of solutions included singular ones, functional solutions and those by infinite series. Many applications were made to mechanics, especially to astronomy and continuous media. In the 19th century: general theory was enriched by development of the un- derstanding of general and particular solutions, and of existence theorems.
    [Show full text]
  • LIE, MARIUS SOPHUS (B
    ndsbv4_L 9/25/07 1:15 PM Page 307 Licklider Lie Technical Documents.” American Documentation 17 (1966): reliable history book on IPTO based on the historical 186–189. documents of the agency that include normally inaccessible With Robert W. Taylor and Evan Herbert. “Computer as a records. Communication Device.” Science & Technology 76 (1968): O’Neill, Judy E. “The Role of ARPA in the Development of the 21–31. ARPANET, 1961–1972.” IEEE Annals of the History of With A. Vezza. “Applications of Information Technology.” Computing 17, no. 4 (1995): 76–81. Proceedings of the IEEE 66 (1978): 1330–1346. Waldrop, M. Mitchell. The Dream Machine: J. C. R. Licklider and the Revolution That Made Computing Personal. New York: OTHER SOURCES Viking, 2001. Biographical history book for general readers Aspray, William, and Arthur L. Norberg. “J. C. R. Licklider.” based on documents and interviews. Author’s interviews with OH 150. 28 October 1988, Cambridge, MA; Charles Louise Licklider and others result in a lively description of Babbage Institute, University of Minnesota, Minneapolis. A Licklider’s personality. scholarly oral history record. “Bolt Beranek and Newman: The First 40 Years.” Special issues, Chigusa Ishikawa Kita IEEE Annals of the History of Computing 27, no. 2 (2005); 28, no. 1 (2006). Some of these papers refer to Licklider’s contribution in the establishment of BBN’s role in the history of computing and networking: Leo Beranek, “BBN’s Earliest Days: Founding a Culture of Engineering Creativity” LIE, MARIUS SOPHUS (b. Nordfjordeide, (27, no. 2: 6–14); John A. Swets, “ABC’s of BBN: From Norway, 17 December 1842; d.
    [Show full text]
  • View This Volume's Front and Back Matter
    Titles in This Series Volume 8 Kare n Hunger Parshall and David £. Rowe The emergenc e o f th e America n mathematica l researc h community , 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moore 1994 7 Hen k J. M. Bos Lectures in the history of mathematic s 1993 6 Smilk a Zdravkovska and Peter L. Duren, Editors Golden years of Moscow mathematic s 1993 5 Georg e W. Mackey The scop e an d histor y o f commutativ e an d noncommutativ e harmoni c analysis 1992 4 Charle s W. McArthur Operations analysis in the U.S. Army Eighth Air Force in World War II 1990 3 Pete r L. Duren, editor, et al. A century of mathematics in America, part III 1989 2 Pete r L. Duren, editor, et al. A century of mathematics in America, part II 1989 1 Pete r L. Duren, editor, et al. A century of mathematics in America, part I 1988 This page intentionally left blank https://doi.org/10.1090/hmath/008 History of Mathematics Volume 8 The Emergence o f the American Mathematical Research Community, 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moor e Karen Hunger Parshall David E. Rowe American Mathematical Societ y London Mathematical Societ y 1991 Mathematics Subject Classification. Primary 01A55 , 01A72, 01A73; Secondary 01A60 , 01A74, 01A80. Photographs o n th e cove r ar e (clockwis e fro m right ) th e Gottinge n Mathematisch e Ges - selschafft, Feli x Klein, J. J. Sylvester, and E. H. Moore.
    [Show full text]
  • Sophus Lie: a Real Giant in Mathematics by Lizhen Ji*
    Sophus Lie: A Real Giant in Mathematics by Lizhen Ji* Abstract. This article presents a brief introduction other. If we treat discrete or finite groups as spe- to the life and work of Sophus Lie, in particular cial (or degenerate, zero-dimensional) Lie groups, his early fruitful interaction and later conflict with then almost every subject in mathematics uses Lie Felix Klein in connection with the Erlangen program, groups. As H. Poincaré told Lie [25] in October 1882, his productive writing collaboration with Friedrich “all of mathematics is a matter of groups.” It is Engel, and the publication and editing of his collected clear that the importance of groups comes from works. their actions. For a list of topics of group actions, see [17]. Lie theory was the creation of Sophus Lie, and Lie Contents is most famous for it. But Lie’s work is broader than this. What else did Lie achieve besides his work in the 1 Introduction ..................... 66 Lie theory? This might not be so well known. The dif- 2 Some General Comments on Lie and His Impact 67 ferential geometer S. S. Chern wrote in 1992 that “Lie 3 A Glimpse of Lie’s Early Academic Life .... 68 was a great mathematician even without Lie groups” 4 A Mature Lie and His Collaboration with Engel 69 [7]. What did and can Chern mean? We will attempt 5 Lie’s Breakdown and a Final Major Result . 71 to give a summary of some major contributions of 6 An Overview of Lie’s Major Works ....... 72 Lie in §6.
    [Show full text]
  • Une Histoire De L'universalité Des Matrices Mathématiques
    Une histoire de l’universalité des matrices mathématiques. Frederic Brechenmacher To cite this version: Frederic Brechenmacher. Une histoire de l’universalité des matrices mathématiques.. 2008. hal- 00339082v1 HAL Id: hal-00339082 https://hal.archives-ouvertes.fr/hal-00339082v1 Preprint submitted on 16 Nov 2008 (v1), last revised 1 Nov 2011 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Une histoire de l’universalité des matrices mathématiques. F. Brechenmacher, prépublication, version du 16/11/2008. Une histoire de l’universalité des matrices mathématiques. Frédéric Brechenmacher (1). Résumé. Cette enquête s’appuie sur des méthodes quantitatives afin de dégager un corpus et une période de référence (1920‐1938) à partir desquels seront mis en œuvre des jeux d’échelles entre local et global, temps court et temps long (1850‐1938). La « théorie des matrices » des années trente manifeste une dynamique d’universalité qui se nourrit de phénomènes multiples et peut s’analyser en termes de passages au global de pratiques locales. Avant qu’un cadre disciplinaire, comme celui de l’algèbre linéaire, permette d’envisager des structures sous tendant les pratiques algébriques, les identités de telles pratiques ne portent le plus souvent pas de significations théoriques mais manifestent des aspects culturels propres à des réseaux dont les identités sont complexes.
    [Show full text]
  • The Equivalence Problem and Signatures of Algebraic Curves
    ABSTRACT RUDDY, MICHAEL GERARD. The Equivalence Problem and Signatures of Algebraic Curves. (Under the direction of Irina Kogan and Cynthia Vinzant). In this thesis we study the application of the differential signature construction to the group equivalence problem for complex algebraic curves under the projective action and its subgroups. Given such an action G, a signature map assigns to a generic algebraic curve an irreducible polynomial, called the signature polynomial, with the property that two curves are G-equivalent if and only if they have the same, up to scaling, signature polynomial. We show that for any action of G, one can construct a pair of rational differential invariants, called classifying invariants, that define a signature map. Given a pair of classifying invariants, elimination algorithms allow one to straightforwardly compute a curve's signature polynomial. In practice, however, these algorithms are computationally intensive, and determination of properties of signature polynomials without their explicit computation is of interest. We derive a formula for the degree of the signature polynomial of a curve in terms of the curve's degree, the size of its symmetry group, and some quantities depending on the choice of classifying invariants. We show that for a given set of classifying invariants and a generic curve of fixed degree, the signature polynomials share the same degree, which is an upper bound. We also show that they share the same monomial support and the same genus. For the projective group and five of its subgroups (the affine, special affine, similarity, Euclidean, and special Euclidean groups), we give sets of classifying invariants and use the degree formula to derive the degree of the signature polynomial for a generic curve as a quadratic function of the curve's degree.
    [Show full text]
  • Lucjan Emil Böttcher (1872‒1937) ‒ the Polish Pioneer of Holomorphic Dynamics
    TECHNICAL TRANSACTIONS CZASOPISMO TECHNICZNE FUNDAMENTAL SCIENCES NAUKI PODSTAWOWE 1-NP/2014 MAŁGORZATA STAWISKA* LUCJAN EMIL BÖTTCHER (1872‒1937) ‒ THE POLISH PIONEER OF HOLOMORPHIC DYNAMICS LUCJAN EMIL BÖTTCHER (1872‒1937) ‒ POLSKI PIONIER DYNAMIKI HOLOMORFICZNEJ Abstract In this article I present Lucjan Emil Böttcher (1872‒1937), a little-known Polish mathematician active in Lwów. I outline his scholarly path and describe briefly his mathematical achievements. In view of later developments in holomorphic dynamics, I argue that, despite some flaws in his work, Böttcher should be regarded not only as a contributor to the area but in fact as one of its founders. Keywords: mathematics of the 19th and 20th century, holomorphic dynamics, iterations of rational functions Streszczenie W artykule tym przedstawiam Lucjana Emila Böttchera (1872‒1937), mało znanego matema- tyka polskiego działającego we Lwowie. Szkicuję jego drogę naukową oraz opisuję pokrótce jego osiągnięcia matematyczne. W świetle późniejszego rozwoju dynamiki holomorficznej do- wodzę, że mimo pewnych wad jego prac należy nie tylko docenić wkład Böttchera w tę dzie- dzinę, ale w istocie zaliczyć go do jej twórców. Słowa kluczowe: matematyka XIX i XX wieku, dynamika holomorficzna, iteracje funkcji wymiernych * Małgorzata Stawiska, Mathematical Reviews, 416 Fourth St., Ann Arbor, MI 48103, USA, email: [email protected]. 234 1. Introduction Holomorphic dynamics – in particular the study of iteration of rational maps on the Riemann sphere – is an active area of current mathematical
    [Show full text]
  • GOLDEN OLDIE 18 on the Three-Dimensional Spaces Which
    GOLDEN OLDIE 18 On the three-dimensional spaces which admit a continuous group of motions Luigi Bianchi Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Terza, Tomo XI, pp. 267–352 (1898). EDITOR’S NOTE This article methodically studies (locally) the symmetry and isometry classes of all 3-dimensional Riemannian manifolds. For each of the possible orbit dimensions 1 and 2 (intransitive actions) and 3 (transitive actions) and for each possible symmetry class of group actions, explicit canonical coordinate expressions are derived for the full Killing vector Lie algebra and the metric by solving the Killing equations. A representative line element is then given parametrizing the isometry classes of a given symmetry type modulo constant conformal transformations, and specializations which admit higher symmetry are studied. For the case of simply transitive 3-dimensional isometry groups, this classification of metrics by symmetry class coincides with the classification into nine isomorphism classes of the isometry groups themselves (Bianchi types I – IX), now known together as the Bianchi classification. This article followed soon after Lie’s classification over the complex numbers of all Lie algebras up to dimension 6 and Killing’s discovery of his famous Killing equations at the end of the nineteenth century. All of Bianchi’s work was well known by the mathematician Luther P. Eisenhart (1876–1965), a professor, chair, dean and important educator in the Princeton University Mathematics Department from 1900 to 1945 [1], who served as a principal source of English language discussion of much of the early work in Riemannian geometry and Lie group theory through his two books Riemannian Geometry (1925) [2] and Continuous Groups of Transformations (1933) [3], which contain numerous references to Bianchi’s work.
    [Show full text]
  • Idempotenten in Groepringen
    FACULTEIT WETENSCHAPPEN EN BIO-INGENIEURSWETENSCHAPPEN DEPARTEMENT WISKUNDE Idempotenten in Groepringen Proefschrift ingediend met het oog op het behalen van de graad van Master in de Wiskunde Inneke Van Gelder Promotor: Prof. Dr. Eric Jespers MEI 2010 FACULTY OF SCIENCE AND BIO-ENGINEERING SCIENCES DEPARTMENT OF MATHEMATICS Idempotents in Group Rings Graduation thesis submitted in partial fulfillment of the requirements for the degree of Master in Mathematics Inneke Van Gelder Promotor: Prof. Dr. Eric Jespers MAY 2010 Voorwoord De lente van 2005, nu exact vijf jaar geleden, een studiekeuze maken bleek plots een niet al te makkelijke opgave. Hoewel een vijf jaar durende studie eerst vreselijk lang leek, weet ik nu dat ik destijds de juiste keuze heb gemaakt. De afgelopen vijf jaren zijn immers voorbij gevlogen en een diploma komt nu duidelijk in zicht. 'Afstuderen' is vaak een synoniem voor 'thesis schrijven': D´eThesis. Een zware taak die ik vol spanning, maar toch ook met de nodige vrees, tegemoet ging. Die vrees was echter nergens voor nodig. Ik houd er een zeer leuke en leerrijke ervaring aan over. Alleen daarom al kan ik voor mezelf stellen dat ik tevreden ben met het resultaat. Nooit had ik gedacht zoveel te kunnen bereiken op enkele maanden tijd. Graag wil ik in het bijzonder mijn promotor bedanken. Eric Jespers, bedankt voor de goede begeleiding, motivatie en de vrijheid die je me gaf. Tevens bedankt dat je me de kans gaf gedurende een volledige week een conferentie bij te wonen. I would like to thank Angel´ del R´ıo.Thank you for your patient explanations, for your interesting lectures and talks and for the everlasting enthousiasm.
    [Show full text]
  • And the Shadow of Camille Jordan)
    Revue d’histoire des mathématiques 17 (2011), p. 273–371 SELF-PORTRAITS WITH EÂVARISTE GALOIS (AND THE SHADOW OF CAMILLE JORDAN) FreÂdeÂric Brechenmacher Abstract. — This paper investigates the collections of 19th century texts in which Evariste Galois’s works were referred to in connection to those of Camille Jordan. Before the 1890s, when object-oriented disciplines developed, most of the papers referring to Galois have underlying them three main networks of texts. These groups of texts were revolving around the works of individuals: Kronecker, Klein, and Dickson. Even though they were mainly active for short periods of no more than a decade, the three networks were based in turn on specific references to the works of Galois that occurred in the course of the Texte reçu le 15 février 2011, accepté le 15 mars 2011. F. Brechenmacher, Univ. Lille Nord de France, 59000 Lille (France), U. Artois, Laboratoire de mathématiques de Lens (EA 2462), rue Jean Souvraz S.P. 18, 62300 Lens (France). Courrier électronique : [email protected] 2000 Mathematics Subject Classification : 01A55, 01A85. Key words and phrases : Galois, Jordan, Hermite, Kronecker, Klein, Dickson, Moore, Traité des substitutions, networks of texts, history of algebra, history of number theory, linear groups, cyclotomy, substitutions, group theory, equations, finite fields, Galois fields. Mots clefs. — Galois, Jordan, Hermite, Kronecker, Klein, Dickson, Moore, Traité des substitutions, réseaux de textes, histoire de l’algèbre, histoire de la théorie des nombres, groups linéaires, cyclotomie, substitutions, théorie des groupes, equations, corps finis. Ce travail a bénéficié d’une aide de l’Agence Nationale de la Recherche: ANR CaaFÉ (ANR-10-JCJC 0101).
    [Show full text]