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

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ISBN 978-3-642-51601-6 ISBN 978-3-642-51599-6 (eBook) DOI 10.1007/978-3-642-51599-6

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