John Charles Fields: A Sketch of His Life and Mathematical Work by Marcus Emmanuel Barnes B.Se., York University 2005 AT HE SIS SUBM ITTED IN PARTIAL F ULFILLMENT OF T HE REQUIREM ENTSFORT HE DEGREE OF l\iIASTEH OF SCIENCE ill the Department of Mathematics CD Marcus Emmanuel Barn es 2007 SIMOr-.: FRASER UNIVERSITY 2007 All rights reserved . This work may not be reproduced ill whol e or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Marcus Emmanuel Barnes Degree: Master of Science Title of Thesis: John Charles Fields: A Sket ch of His Life and Mathe­ matical Vv'ork Examining Committee: Dr. J amie Mulholland Chair Dr. Tom Archibald Senior Supervisor Dr. Glen Van Brummelen Supervisory Committee Dr. J ason Bell Internal Examiner Date of Defense: November 29, 2007 11 SIMON FRASER UNIVERSITY LIBRARY Declaration of Partial Copyright Licence The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection (currently available to the public at the "Institutional Repository" link of the SFU Library website -cwww.Ilb.stu.ca» at: <http://ir.lib.sfu.calhandle/1892/112>) and, without changing the content, to translate the thesis/project or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without the author's written permission. Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright Licence. While licensing SFU to permit the above uses, the author retains copyright in the thesis, project or extended essays, including the right to change the work for subsequent purposes, including editing and publishing the work in whole or in part, and licensing other parties, as the author may desire. The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive. Simon Fraser University Library Burnaby, BC, Canada Revised: Fall 2007 Abstract Every four years at. the International Congress of Mat hematicians the prestigious Fields medals, the mathematical equivalent, of a Nobel prize, are awarded . The fol­ lowing question is often asked: who was Fields and what did he do mathematically? This question will be addressed by sketching the life and mathematical work of John Charles Fields (1863 - 1932) , th e Canadian mathematician who helped establish the awards and after whom the medals are named. }II Dedication To T he.Tortoise - keep on plugging a.long... lV Acknowledgments Many peo ple helped me through the 'writ ing of this thesis - to many to list here. You know who you are and I thank you very much for all the help you have given me. However, certain indi viduals do need specific menti on. First, I would like to thank my supervisor, Prof. Tom Archibald. Wi th out his guidance, this docu ment. would never have been completed. He also aided me in several tasks of translating German into English , including the reviews qu oted in Chap ter 5 of Fields ' work. Also, I would like to mention two memb ers of my family who have been my support "on th e ground", so to speak, here in Vancouver during my st udies: my mother, Heather, and my brother, St even. I would also like t o ment ion the staff in the department of mathematics at Simon Fraser University: they were always kind and helpfu l throughout my st udies th ere. I would further like to acknowledge the help of th e staff at the University of Toronto Archives an d J ames Stimpert of the Milton J . Eisenhower Library at John s Hopkins University 'wit h help findin g material related to Fields. The Milton J . Eisenhower library kindl y let me use the portrait of Fields in their collect ion. Lastly, I would like to thank my examiners for their comments on a draft version of this document. v Contents Approval . 11 Abstract III Dedication . IV Acknowledgments . v Contents . VI List. of Figures . V lll 1 Introduction 2 I A Biographical Sketch of John Charles Fields 6 2 Fields' Youth and Education . 7 2.1 Field 's Youth .. .. .. .... 7 2.2 Graduate St.udy a t. Johns Hopkins University 8 2.3 Post Doctoral Studies in Eur ope. 12 3 Professor Fields: 1900-1932 . 19 II A Sketch of John Charles Fields' M athematical Con­ tributions 26 4 Skill in manipulat ions . .. .... .. 29 4.1 Fields' PhD Thesis . 30 4.1.1 Finite SYmbolic Solutions . 32 .. " . , . d°,lj _ .m 4.1.2 F1l11 te Soluti ons of d:r:; - :r y . 36 4.1.3 Solut.ions by Definite Int.egrals . 39 4.2 Some of Fields' Other Earlv Mathematical Work . 45 VI 4.2.1 A Proof of the Theorem: The Equat ion f (;:;) = () Has a Root. 'Where f( ::; ) is any Holomorphic Functi on of z 45 4.2.2 A Proof of the Elliptic-Function Addition-Theorem 47 4.2.3 Expressions for Bernoulli 's and Euler 's Numbers . 50 4.2.4 Summary of Fields' Early Mathematical Work . 52 5 Fields ' Work in Algebr aic Function Theory . 54 5.1 Element.s of Algebraic Function Theory 56 5.2 Fields' Theory of Algebraic Functions . 64 6 Conclusion: The Legacy of J. C. Fields 71 Appendi x 76 A ANot e on Fields' Berlin Notebo oks. 76 B The Fields Medal 76 Bibliography 78 Vll List of Figures Fig. 1: John Charles F ields (1863-1932). © Ferdinand Hamburger Archives of The Johns Hopkins Un iversity . .. .. .. 1 Fig. 2.1: A drawing in one of Fields' notebooks showing branch cuts styl­ ized as a bug. UTA (Univers ity of Toronto Archives), J . C. Fi elds, BI972-0024. 16 Fi g. 2.2: A drawing in one of Fi elds' notebooks showing branch points sty lized as a. map of the Berlin subway stops in the 1890s. UTA (Univers ity of Toronto Archives) , J. C. Fields, BI972-0024. 17 Fi g. 2.3: A page from Fields' notebook of Hensel's lectures on algebraic functions for 1897-98 showing a diagram of a Newton polygon for an algebraic function. UTA (Universi ty of Toronto Archives) , J . C. Fields, BI972-0024. ... .. .. 18 F ig. 3.1: J . C. F ields' gravestone:"John Ch arles Fields , Born May 14, 1863, Died Au gust 9, 1932." © Carl Riehm . Used with per- m ISSIOn . 25 Fig. 4.1: F igure from ([Fie86b], 178). 46 Fig. 5.1: An depiction ofthe Riemann Surface for the square root fun cti on . From [AG29, 27].. .. ... .. .. ... 62 Fi g. 6.1: T he Fields Medal given to Maxim Kontsevich in 1998 in Berlin. © Internat. ional Mathematical Union. Used with permission .. 77 V111 1 Figure 1: J ohn Charles Fields (1863-1932). © Ferdinand Hamburger Archives of The Johns Hopk ins University Chapter 1 Introduction Many a mathematician has heard of t.he foremost. award in the mathematical sciences, one often said to be the "Nobel priz e" for mathematics, namely, the Fields med al. Few people, mathematicians included , have any idea of exactly who th e mysterious Fields was. It may come as a sur prise to some that thi s medal was named after a modest Ca nadian professor of mathematics a t. the University of Toron to, J ohn Charles Fields (1863-1932) . Unfortunately, little is known abo ut Fields, except what appears in the stan­ dard biographical account of his life, an obituary by J . L. Synge [Syn33] . Much of Fields' correspondence must be presum ed missing or destr oyed after it was distribu ted upon his deat h in 1932 (many of his pos sessions were sent to his brother in Cali for­ nia [Fie33]). However, the historian is fortunat e on some levels. Fields left many notebooks of lectures he either attended or copied from others from a postdoctoral sojourn in Germany in 1890s, spending most of this time in Berlin. There are also files of the minutes of the Toronto International Congress of Mathematics meetin g committee held in the archives of the International Mathematical Union, which have allowed scholars like Henry S. Tropp in [Tro76 ] and OU i Lehto in [Leh9 -'] to det ail the early history of the Fields medals and Fields' role in the organi zation of the 1924 2 CHAPTER 1. INTRODUCTION 3 Intern ational Congress of Mathematicians, held in Toronto. For a look at modern mathematics in the light of the Fields medals, see Michael Monastyrskys book on t he subject [IVlon97] . Other lit erature on Fields includes an entry on Fields by Tropp in the Dictionary of Scientific Biography [Tr074].T here is also a small number of general interest accounts of Fields' life and his role in the creation of the Fields medals, and these include articles by C.