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HISTORY OF MATHEMATICS • VOLUME 32 Episodes in the History of Modern Algebra (1800–1950) Jeremy J. Gray Karen Hunger Parshall Editors American Mathematical Society • London Mathematical Society Episodes in the History of Modern Algebra (1800–1950) https://doi.org/10.1090/hmath/032 HISTORY OF MATHEMATICS v VOLUME 32 Episodes in the History of Modern Algebra (1800–1950) Jeremy J. Gray Karen Hunger Parshall Editors Editorial Board American Mathematical Society London Mathematical Society Joseph W. Dauben Alex D. D. Craik Peter Duren Jeremy J. Gray Karen Parshall, Chair Robin Wilson, Chair MichaelI.Rosen 2000 Mathematics Subject Classification. Primary 01A55, 01A60, 01A70, 01A72, 01A73, 01A74, 01A80. For additional information and updates on this book, visit www.ams.org/bookpages/hmath-32 Library of Congress Cataloging-in-Publication Data Episodes in the history of modern algebra (1800–1950) / Jeremy J. Gray and Karen Hunger Parshall, editors. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-8218-4343-7 (alk. paper) ISBN-10: 0-8218-4343-5 (alk. paper) 1. Algebra—History. I. Gray, Jeremy, 1947– II. Parshall, Karen Hunger, 1955– QA151.E65 2007 512.009—dc22 2007060683 AMS softcover ISBN: 978-0-8218-6904-8 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2007 by the American Mathematical Society. All rights reserved. Reprinted in softcover by the American Mathematical Society, 2011. Printed in the United States of America. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. The London Mathematical Society is incorporated under Royal Charter and is registered with the Charity Commissioners. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 Contents Acknowledgments 1 Chapter 1. Introduction Jeremy J. Gray and Karen Hunger Parshall 3 Algebra: What? When? Where? 3 Episodes in the History of Modern Algebra 5 Concluding Remarks 10 References 10 Chapter 2. Babbage and French Id´eologie: Functional Equations, Language, and the Analytical Method Eduardo L. Ortiz 13 Introduction 13 Speculation on the Origin of Languages 14 Senses, Languages, and the Elaboration of a Theory of Signs 16 The Position of Grammar 17 On the Language of Calculation 18 Babbage and a “Language” for the Solution of Functional Equations 19 Babbage’s Notation 21 Babbage’s Treatment of Functional Equations 23 Babbage and First-Order Functional Equations in One Variable 26 The Aftermath of Condillac in France: The Beginning of a Discussion 28 De G´erando’s Critique of Condillac: A Turning Point in Id´eologie 31 De G´erando’s Theory of Signs and Its Mathematical Implications 33 Babbage and de G´erando’s Views on Signs 36 Destutt de Tracy’s El´´ emens d’id´eologie 37 Destutt de Tracy’s Views on Artificial Languages 39 Final Remarks 40 References 42 Chapter 3. “Very Full of Symbols”: Duncan F. Gregory, the Calculus of Operations, and the Cambridge Mathematical Journal Sloan Evans Despeaux 49 Introduction 49 The Establishment of a “Proper Channel” for the Research of Junior British Mathematicians 50 The Calculus of Operations before the Cambridge Mathematical Journal 52 The Revival: The Introduction of the Calculus of Operations into the Cambridge Mathematical Journal 54 v vi CONTENTS The Conversation Begins: The Adoption of the Calculus of Operations by Contributors to the Cambridge Mathematical Journal 60 Conclusion 67 References 69 Chapter 4. Divisibility Theories in the Early History of Commutative Algebra and the Foundations of Algebraic Geometry Olaf Neumann 73 On Some Developments Rooted in the Eighteenth Century 74 Developments Inspired by Gauss 77 From Kummer to Zolotarev 84 Complex Analytic and Algebraic Functions 87 Kronecker’s Modular Systems 89 The Work of David Hilbert and Emanuel Lasker 94 The Work of Emmy Noether and Her Successors 97 Concluding Remarks 99 References 100 Chapter 5. Kronecker’s Fundamental Theorem of General Arithmetic Harold M. Edwards 107 Introduction 107 Statement and Proof of Kronecker’s Theorem 109 Conclusions 114 References 115 Chapter 6. Developments in the Theory of Algebras over Number Fields: A New Foundation for the Hasse Norm Residue Symbol and New Approaches to Both the Artin Reciprocity Law and Class Field Theory Gunther¨ Frei 117 Introduction 117 The Beginnings of Structure Theory 118 Wedderburn’s General Structure Theorems 121 Hurwitz and the Arithmetic of Quaternions 122 The Structure of Skew Fields: Connections with Algebraic Number Theory 124 The Theory of Semisimple Algebras 127 The Local Theory and the Theorem of Brauer-Hasse-Noether 133 The New Norm Residue Symbol and New Approaches to Both the Reciprocity Law and Class Field Theory 135 Summary and Conclusions 141 References 143 Chapter 7. Minkowski, Hensel, and Hasse: On The Beginnings of the Local-Global Principle Joachim Schwermer 153 Introduction 153 Toward an Arithmetic Theory of Quadratic Forms 155 Mathematical Digression: Quadratic Forms over Rings 158 Hermann Minkowski’s Early Work 159 Hermann Minkowski’s Letter to Adolf Hurwitz in 1890 161 CONTENTS vii Hensel’s p-adic Numbers: Series, Expansions, or Numbers as Functions 162 Mathematical Digression: Valuations and p-adic Fields 164 Hasse’s Thesis and Habilitationsschrift 165 From the Small to the Large, or a Local-Global Principle 167 A Talk by Hasse in K¨onigsberg in 1936 168 Conclusion 170 Appendix: Helmut Hasse to Hermann Weyl, 15 December, 1931 171 References 173 Chapter 8. Research in Algebra at the University of Chicago: Leonard Eugene Dickson and A. Adrian Albert Della Dumbaugh Fenster 179 Introduction 179 Leonard Dickson: Student 179 Leonard Dickson: University of Chicago Faculty Member 180 A. Adrian Albert and the Classification of Division Algebras 181 A. Adrian Albert: Professional Overview 188 Conclusions 192 References 194 Chapter 9. Emmy Noether’s 1932 ICM Lecture on Noncommutative Methods in Algebraic Number Theory Charles W. Curtis 199 Introduction 199 Brauer’s Factor Sets, the Brauer Group, and Crossed Products 202 Cyclic Algebras and the Albert-Brauer-Hasse-Noether Theorem 206 The Principal Genus Theorem 210 Applications to Algebraic Number Theory by Hasse and Chevalley 212 Conclusion 215 References 217 Chapter 10. From Algebra (1895) to Moderne Algebra (1930): Changing Conceptions of a Discipline–A Guided Tour Using the Jahrbuch uber¨ die Fortschritte der Mathematik Leo Corry 221 Introduction 221 The Jahrbuchuber ¨ die Fortschritte der Mathematik 225 Algebra by the Turn of the Century: The Jahrbuch in 1900 227 Some Tentative Changes: 1905–1915 229 Transactions of the American Mathematical Society: 1910 232 The Jahrbuch after 1916 233 Concluding Remarks 237 References 239 Chapter 11. A Historical Sketch of B. L. van der Waerden’s Work in Algebraic Geometry: 1926–1946 Norbert Schappacher 245 Introduction 245 1925: Algebraizing Algebraic Geometry `alaEmmy Noether 250 1927–1932: Forays into Intersection Theory 256 viii CONTENTS 1933–1939: When in Rome . ? 264 1933–1946: The Construction Site of Algebraic Geometry 272 Appendix: Extract from a Letter from Hasse to Severi 276 References 278 Chapter 12. On the Arithmetization of Algebraic Geometry Silke Slembek 285 Introduction 285 Earlier Rewritings of Algebraic Geometry 286 The Mathematical Situation: Why Another Rewriting Seemed Necessary 289 Imitating Geometry with Modern Algebra and Arithmetic Ideal Theory 292 Leaving the Beaten Path 294 The Arithmetic Proof 296 Conclusion 296 References 298 Chapter 13. The Rising Sea: Grothendieck on Simplicity and Generality Colin McLarty 301 The Weil Conjectures 303 Abelian Categories 305 The Larger Vision 311 Anticipations of Schemes 313 Schemes in Paris 315 Schemes in Grothendieck 317 Toward the S´eminaire de G´eom´etrie Alg´ebrique 321 References 322 Index 327 Index Abel, Niels Henrik, 4, 76 internationalization of, 192 as father of the theory of algebraic func- Kronecker’s views on the foundations of, tions, 87 107–109 on the solution of algebraic equations, 111 origin of the term, 3 Abelian categories, 305–311 Peacock’s views on, 58–59 Abelian functions research in around 1800, 74 Riemann’s work on, 87–88 the role of the calculus of operations in theory of, 87–88 the development of in Britain, 68 al-Khw¯arizm¯ı, 3 Algebraic functions Albert, A. Adrian Abel as father of, 87 as an adviser, 190 Dedekind’s work on, 88–89 broader community interests of, 189–190 E. Noether’s work on, 89 career of, 188–192 Kronecker’s work on, 88 correspondence of with Hasse, 184–187 theory of, 87–89 early education of, 181–183 Weber’s work on, 88–89 joint paper
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