HISTORY OF MATHEMATICS
CONTENTS CHAPTER 1 INTRODUCTION 1 CHAPTER 2 BABBAGE AND FRENCH ID OLOGIE FUNCTIONAL EQUATIONS LANGUAGE AND THE ANALYTICAL METHOD 11 CHAPTER 3 VERY FULL OF SYMBOLS DUNCAN F GREGORY THE CALCULUS OF OPERATIONS AND THE CAMBRIDGE MATHEMATICAL JOURNAL 47 CHAPTER 4 DIVISIBILITY THEORIES IN THE EARLY HISTORY OF COMMUTATIVE ALGEBRA AND THE FOUNDATIONS OF ALGEBRAIC GEOMETRY 71 CHAPTER 5 KRONECKER S FUNDAMENTAL THEOREM OF GENERAL ARITHMETIC 105 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 115 CHAPTER 7 MINKOWSKI HENSEL AND HASSE ON THE BEGINNINGS OF THE LOCAL GLOBAL PRINCIPLE 151 CHAPTER 8 RESEARCH IN ALGEBRA AT THE UNIVERSITY OF CHICAGO LEONARD EUGENE DICKSON AND A ADRIAN ALBERT 177 CHAPTER 9 EMMY NOETHER S 1932 ICM LECTURE ON NONCOMMUTATIVE METHODS IN ALGEBRAIC NUMBER THEOR 197 CHAPTER 10 FROM ALGEBRA 1895 TO MODERNE ALGEBRA 1930 CHANGING CONCEPTIONS OF A DISCIPLINE A GUIDED TOUR USING THE JAHRBUCH BER DIE FORTSCHRITTE DER MATHEMATIK 219 CHAPTER 11 A HISTORICAL SKETCH OF B L VAN DER WAERDEN S WORK IN ALGEBRAIC GEOMETRY 1926 1946 243 CHAPTER 12 ON THE ARITHMETIZATION OF ALGEBRAIC GEOMETRY 283 CHAPTER 1 INTRODUCTION ALGEBRA WHAT WHEN WHERE IT IS AN INTERESTING AND DI CULT PROBLEM EVEN TO DE NE ALGEBRA TO BEGIN TO GIVE IT A HISTORY PRESENTS YET ANOTHER CHALLENGE SOME HISTORIANS OF MATHEMATICS IDENTIFY THE ABSTRACT AND GENERAL FEATURES OF MESOPOTAMIAN MATHEMATICS WITH AL GEBRA AND SO CONSIDER IT THE OLDEST BRANCH OF WRITTEN MATHEMATICS DATING FROM AT LEAST 2000 B C E 1 OTHERS SEE ITS ORIGINS AROUND 150 C E IN DIOPHANTUS S ARITH METICA WITH ITS SOLUTIONS OF GENERAL INDETERMINATE PROBLEMS KLEIN 1968 STILL OTHERS ND ITS MORE MODERN ROOTS IN MEDIEVAL ISLAM NOTING THAT THE VERY WORD ORIGINATED ONLY IN THE NINTH CENTURY IN AL KHW ARIZM S TEXT ENTITLED AL KIT AB AL MUKHTASAR F HIS AB AL JABR WA L MUQ ABALA OR THE COMPENDIOUS BOOK ON CALCULATION BY COMPLETION AND BALANCING THERE AL JABR WHICH WAS ULTIMATELY LATINIZED INTO ALGEBRA TRANSLATES AS COMPLETION OR RESTORATION IT REFERRED TO ADDING THE SAME THING TO BOTH SIDES OF AN EQUATION OR RESTORING ALL OF THE TERMS TO A STAN DARD FORM AND AL KHW ARIZM HIMSELF REGARDED IT AS DI ERENT FROM ALTHOUGH NOT UNRELATED TO THE KIND OF GEOMETRICAL PROCEDURES THAT EUCLID HAD UTILIZED IN THE ELEMENTS 300 B C E AND THAT HAD HELD SWAY OVER THE COURSE OF THE INTERVEN ING MILLENNIUM VAN DER WAERDEN 1985 THOSE HISTORIANS WHO INSIST ON ALGEBRA AS SOMETHING DISTINCT FROM GEOMETRY HOWEVER SEE ITS EMERGENCE IN THE SIXTEENTH CENTURY EUROPEAN TEXTS OF MATHEMATICIANS SUCH AS GIROLAMO CARDANO NICCOL O TARTAGLIA AND RAFAEL BOMBELLI ON CUBIC AND QUARTIC EQUATIONS AS WELL AS ON THE INTRODUCTION OF COMPLEX NUMBERS 2 WHILE OTHERS REQUIRE THE PRESENCE OF THE CHAR ACTERISTIC FEATURES OF HIGH SCHOOL ALGEBRA TODAY NAMELY THE FAMILIAR SIGNS FOR THE ARITHMETIC OPERATIONS AND THE USE OF LETTERS FOR CONSTANTS AND UNKNOWNS FOR THE LATTER ALGEBRA CAME ABOUT IN THE LATE SIXTEENTH AND SEVENTEENTH CENTURIES IN THE WORK OF FRAN COIS VI ETE THOMAS HARRIOT PIERRE DE FERMAT AND REN E DESCARTES 3 FINALLY MATHEMATICIANS TEND TO SHARE YET ANOTHER VIEW OF ALGEBRA S ORIGINS FOR 1FOR A VERY SOPHISTICATED AND HIGHLY NUANCED INTERPRETATION OF BABYLONIAN ALGEBRA SEE H YRUP 2002 MANY HISTORIES OF ALGEBRA BEGIN THEIR STORY IN ANCIENT MESOPOTAMIA SEE FOR EX AMPLE VAN DER WAERDEN 1983 SCHOLZ 1990 SESIANO 1999 AND BASHMAKOVA AND SMIRNOVA 2000 2SEE FOR EXAMPLE CARDANO 1545 1968 FOR A SENSE OF THIS SIXTEENTH CENTURY ITALIAN WORK 3FOR A SENSE OF THE SEVENTEENTH CENTURY SCENE SEE BOS 2001 IN NO VY 1973 THE HISTORY OF ALGEBRA BEGINS SOME TIME AFTER 1770 1 THEM ALGEBRA IS MODERN ALGEBRA NAMELY THE SUBJECT THAT WAS PUT INTO BOOK FORM AND GIVEN ITS NAME BY BARTEL VAN DER WAERDEN BEGINNING IN 1930 4 ALL OF THESE DE NITIONS HAVE SOMETHING TO RECOMMEND THEM AND INDEED THERE IS A DEGREE OF CONTINUITY BETWEEN THESE TYPES OF ALGEBRA THAT SUGGESTS A COHERENCE TO THE TOPIC ACROSS THE CENTURIES THE ANCIENT MESOPOTAMIANS POSSESSED SYSTEMATIC METHODS FOR SOLVING NUMERICAL QUESTIONS ABOUT NUMBERS AND THEIR RECIPROCALS AND ABOUT NUMBERS AND SQUARES INTEREST IN THIS TOPIC PASSED TO MEDIEVAL ISLAMIC MATH EMATICIANS AND EVENTUALLY FROM THEM TO THE RENAISSANCE ITALIANS WHEN IN THE LATE EIGHTEENTH AND EARLY NINETEENTH CENTURIES THE QUINTIC EQUATION PROVED A STUMBLING BLOCK TO THE PRODUCTION OF ALGORITHMS OR FORMULAS FOR SOLVING POLYNOMIAL EQUATIONS OF SUCCESSIVE DEGREES A SERIES OF MATHEMATICIANS TOOK UP STRUCTURAL INVESTIGATIONS OF THE PROBLEM ALTHOUGH THE LIST OF STRUCTURALLY MINDED ALGEBRAISTS COULD JUSTI ABLY BEGIN WITH HARRIOT 5 IT MOST CERTAINLY INCLUDES THE NAMES OF THE EIGHTEENTH AND EARLY NINETEENTH CENTURY MATHEMATICIANS ALEXANDRE VANDERMONDE JOSEPH LOUIS LAGRANGE NIELS HENRIK ABEL AND EVARISTE GALOIS 6 IF THE SOLUTION OF EQUATIONS HAS LONG ENGAGED MATHEMATICAL MINDS SO TOO HAS THE PROBLEM OF PROVIDING INTEGER AND INDEED NATURAL NUMBER SOLUTIONS TO MATH EMATICAL QUESTIONS WHETHER THIS BEGAN AND CONTINUED IN A CULTURE THAT FOUND IT NATURAL TO RESTRICT THE RANGE OF SOLUTIONS IN THIS FASHION SO NATURAL IT WENT WITHOUT COMMENT OR WHETHER IT WAS A DELIBERATELY IMPOSED RESTRICTION A LONG STANDING TRADITION CONNECTS ALGEBRA WITH NUMBER THEORY AND SPECI CALLY WITH THE BRANCH TODAY CALLED ALGEBRAIC NUMBER THEORY FERMAT MAY HAVE LABORED IN VAIN TO INTEREST HIS CONTEMPORARIES IN THE SUBJECT AND SUCH WAS HIS MELANCHOLY OPINION BUT WITH LEONHARD EULER AND LAGRANGE THE SUBJECT BURGEONED IN THE EIGHTEENTH CENTURY AND BECAME ONE OF THE KEY AREAS OF INTEREST TO THE GERMAN MATHEMATICAL COMMUNITY IN THE NINETEENTH 7 ALGEBRA HAS ALSO ESTABLISHED A CLOSE CONNECTION WITH GEOMETRY WHILE SOME HAVE ARGUED THAT EUCLID WAS DOING GEOMETRICAL ALGEBRA AS EARLY AS 300 B C E THE MODERN CONNECTION BETWEEN THE TWO ELDS IS MUCH MORE RECENT NOT EXTENDING BACK CHRONOLOGICALLY MUCH BEFORE THE TIME OF FERMAT AND DESCARTES 8 THIS HOW EVER BECAME A VITAL CONNECTION WITH EACH DISCIPLINE AT TIMES TURNING TO THE OTHER FOR CENTRAL INSIGHTS AND METHODS EULER WAS THE RST TO DE NE THE CONIC SECTIONS ALGEBRAICALLY RENDERING IT OBVIOUS THAT THE SAME B2 4AC TEST THAT DISTINGUISHED ELLIPSES FROM HYPERBOLAS DISTINGUISHED THE REAL AND COMPLEX ROOTS OF A POLYNOMIAL EQUATION IN THE NINETEENTH CENTURY MATHEMATICIANS FROM GEORGE BOOLE ARTHUR CAYLEY AND JAMES JOSEPH SYLVESTER IN GREAT BRITAIN TO CHARLES HERMITE PAUL GOR DAN AND DAVID HILBERT ON THE CONTINENT PURSUED THIS CLUE ENERGETICALLY AND IN SO DOING DEVELOPED THE NEW ELD OF INVARIANT THEORY 9 ALGEBRAIC AND ESPECIALLY PROJECTIVE GEOMETRY WAS ONE HALF OF A SUBJECT INVARIANT THEORY THE OTHER AND SO 4ON THE RISE OF THE STRUCTURAL APPROACH TO ALGEBRA THAT CULMINATED IN SOME SENSE IN VAN DER WAERDEN S TEXT SEE CORRY 1996 5ON HARRIOT S MATHEMATICAL WORK SEE STEDALL 2003 6FOR AN ANALYSIS OF THESE DEVELOPMENTS SEE FOR EXAMPLE EDWARDS 1984 7SEE EULER 1770 1771 1984 WITH ITS ADDITIONS BY LAGRANGE 8THE DEBATE OVER GEOMETRICAL ALGEBRA ARED FAMOUSLY IN AN EXCHANGE BETWEEN BARTEL VAN DER WAERDEN AND SABATEI UNGURU ON THE PAGES OF THE ARCHIVE FOR HISTORY OF EXACT SCIENCES SEE UNGURU 1975 AND VAN DER WAERDEN 1976 9ON THESE DEVELOPMENTS SEE FOR EXAMPLE PARSHALL 1989 HISTORY OF MATHEMATICS 2 BEGAN AN AMICABLE AND LONG RUNNING FAMILY DISPUTE BETWEEN THOSE WHO STRESSED GEOMETRY AND THOSE WHO EMPHASIZED ALGEBRA ALGEBRA EVEN HAS A CLAIM TO BEING IF NOT THE CORE OF MATHEMATICS THEN AT LEAST ITS LANGUAGE MOST PAPERS IN ANY BRANCH OF MATHEMATICS HAVE AN ALGEBRAIC CAST TO THEM DETERMINED BY THE VERY SYMBOLISM USED EVEN IF THE INTELLECTUAL DI CULTIES ARE IN ANOTHER ELD THIS FACT LED SOME EIGHTEENTH CENTURY PHILOSOPHERS TO REGARD MATHEMATICS THAT IS ALGEBRAIC MATHEMATICS AS THE BEST LANGUAGE BECAUSE IT GEN ERATED ARGUMENTS HAVING THE HIGHEST DEGREE OF CERTAINTY TWO HUNDRED YEARS LATER MATHEMATICIANS TURNED TO SYNTAX AND FORMAL LANGUAGES TO SEEK ASSURANCES ABOUT THE NATURE OF MATHEMATICS WHILE LINGUISTS SUCH AS NOAM CHOMSKY DE NED LANGUAGE ITSELF AS A PURELY SYNTACTIC OBJECT EXPRESSED IN MATHEMATICAL TERMS ALGEBRAS LAN GUAGES COMPLEXITY AND ISSUES IN COMPUTATION AND THE THEORY OF ALGORITHMS ARE CENTRAL TO SUCH TOPICS AS THE P VERSUS NP QUESTION ALGEBRA THEN HAS NOT ONLY AN EVER EVOLVING AND DISTINCT IDENTITY BUT ALSO CLOSE CONNECTIONS TO ARITHMETIC AND NUMBER THEORY TO GEOMETRY AND TO TOPICS IN LINGUISTICS AND COMPUTER SCIENCE IT IS A VAST AND VASTLY COMPLICATED AREA TO TRY TO PIN DOWN FROM AN HISTORICAL POINT OF VIEW AS NOTED IN THE ACKNOWLEDGMENTS THIS BOOK GREW OUT OF A CONFERENCE HOSTED BY THE MATHEMATICAL SCIENCES RESEARCH INSTITUTE BERKELEY IN 2003 IT THUS EXPLORES A NUMBER OF EPISODES IN THE MODERN PERIOD OF ALGEBRA S DEVELOPMENT SPECI CALLY EPISODES THAT STRETCH FROM THE TIME OF THE FRENCH REVOLUTION IN THE LATE EIGHTEENTH CENTURY TO THE ARRIVAL OF MODERN ALGEBRA IN THE EARLY TWENTIETH CENTURY TO THE ADVENT OF CATEGORY THEORY AS AN ALGEBRAIC TOOL IN THE MID TWENTIETH CENTURY EPISODES IN THE HISTORY OF MODERN ALGEBRA THE VOLUME OPENS WITH A CHAPTER BY EDUARDO ORTIZ ON BABBAGE AND FRENCH ID EOLOGIE FUNCTIONAL EQUATIONS LANGUAGE AND THE ANALYTICAL METHOD THAT EX PLORES EIGHTEENTH AND EARLY NINETEENTH CENTURY CONCEPTIONS OF MATHEMATICS AS A LANGUAGE AS ORTIZ SHOWS THIS WAS A PARTICULAR CONCERN OF THE ABB E DE CONDILLAC WHO ADVOCATED INVENTING AND ADAPTING LANGUAGES IN THE LIGHT OF THE EXPERIENCE PRO VIDED BY VARIOUS SCIENCES TAKING ALGEBRA AS A MODEL THIS REVIVED A DEBATE THAT HAD OURISHED IN THE SEVENTEENTH CENTURY ABOUT THE RELATIONSHIP BETWEEN LOGIC AND GRAMMAR BUT CONDILLAC REVERSED THE OLD VIEW AND ARGUED THAT MASTERY OF GRAMMAR SHOULD COME BEFORE THE STUDY OF LOGIC AS THIS MAY SUGGEST CONDILLAC WAS INTER ESTED IN THE IDEA THAT MENTAL PROCESSES INVOLVED IN UNDERSTANDING SUCH AS ANALOGY AND ASSOCIATION OF IDEAS SHOULD FEATURE MORE PROMINENTLY IN THE LANGUAGE IN WHICH THOUGHT IS EXPRESSED CHARLES BABBAGE READ CONDILLAC S WORK AS A YOUNG MAN AND APPLIED IT TO HIS OWN RESEARCH ON FUNCTIONAL EQUATIONS THE IDEA OF ANALOGY IN A CONTEXT THAT FOR BABBAGE WAS MUCH MORE ALGEBRAIC THAN ANALYTIC STRUCK HIM FORCEFULLY AND HE DEVISED A METHOD FOR CONSTRUCTING GENERAL SOLUTIONS WHEN SPECIAL CASES ARE GIVEN THE METHOD ORTIZ ARGUES SEEMS CAPABLE OF FURTHER USE EVEN TODAY BUT HAS NEVER BEFORE BEEN ADEQUATELY DESCRIBED IN THE HISTORICAL LITERATURE THE THEMES OF LANGUAGE THE OPERATIONAL CALCULUS AND BROADER FORCES AT PLAY IN THE HISTORY OF MATHEMATICS ARE CONTINUED IN THE CHAPTER BY SLOAN DESPEAUX ON DUN CAN GREGORY AND THE CAMBRIDGE MATHEMATICAL JOURNAL THE EMERGING GENERATION OF BRITISH MATHEMATICIANS IN THE 1820S AND 1830S WAS LARGELY INDI ERENT TO THE STRIC TURES OF AUGUSTIN LOUIS CAUCHY AND OTHER CONTINENTAL EUROPEANS ABOUT THE LACK OF RIGOR IN THE FORMAL METHODS OF THE OPERATIONAL CALCULUS THEY APPLIED IT CAVALIERLY HISTORY OF MATHEMATICS 3 BUT WITH SOME SUCCESS TO VARIOUS KINDS OF DI ERENTIAL EQUATIONS AND WHILE THEY HAD TO ADMIT THEY COULD NOT MAKE IT WORK FOR EQUATIONS WITH VARIABLE COE CIENTS THEY WERE NONETHELESS LED TO INTERESTING RE ECTIONS ABOUT ALGEBRAIC PROCESSES MOREOVER ALTHOUGH THEY DID NOT SUCCEED IN THEIR GOAL OF CREATING A SINGLE UNI ED ELD THAT EMBRACED POLYNOMIAL EQUATIONS DI ERENTIAL EQUATIONS AND FUNCTIONAL EQUATIONS THEY DID CREATE AND SUSTAIN A JOURNAL FOR MATHEMATICS IN BRITAIN THAT SERVED AS A KEY LINE OF COMMUNICATION FOR A DEVELOPING COMMUNITY OF BRITISH MATHEMATICIANS AMONG THE JOURNAL S NOTABLE PUBLICATIONS WAS GEORGE BOOLE S WORK ON LINEAR DI ER ENTIAL EQUATIONS WITH CONSTANT COE CIENTS A PAPER WHICH HIGHLIGHTED NOT ONLY THE POTENTIAL STRENGTH OF THE CAMBRIDGE STYLE FORMAL METHODS BUT ALSO THE JOURNAL S E ECTIVENESS IN CONVEYING NEW MATHEMATICAL IDEAS TO A SELF TAUGHT BOOLE WITH NO REAL CAMBRIDGE CONNECTIONS OTHER THAN ACCESS TO ITS PAGES THE EPISODES IN THE HISTORY OF MODERN ALGEBRA THAT ORTIZ AND DESPEAUX DETAIL SEEM FAR FROM THE MODERN ALGEBRA THAT HAS OFTEN BEEN SEEN AS HAVING SPRUNG WITH ALL THE INEVITABILITY OF A GREAT INVENTION IN THE EARLY DECADES OF THE TWENTIETH CENTURY FROM THE GROUP AROUND EMMY NOETHER IT IS AS HARD FOR MATHEMATICIANS TO IMAGINE THE TWENTIETH CENTURY WITHOUT IT AS IT IS TO IMAGINE THE WORLD WITHOUT HOLLYWOOD AND THIS HAS HINDERED HISTORIANS ABILITY TO DESCRIBE THE TRANSFORMATION FROM THE OLDER ALGEBRA TO THE NEW IT HAS THUS SEEMED EASIER TO LOOK AT THE PRECURSORS IN THE NOETHER LINE SUCH AS RICHARD DEDEKIND THAN TO EXPLORE THE WORK OF THOSE WHO STAND OUTSIDE IT THE NEXT TWO CHAPTERS HERE HOWEVER TAKE ON THAT HARDER TASK OLAF NEUMANN TRACES THE CONCEPT OF DIVISIBILITY FROM ITS ORIGINS IN AND EVEN BEFORE THE EIGHTEENTH CENTURY TO THE TWENTIETH CENTURY WORK OF EMMY NOETHER AND WOLFGANG KRULL THE IDEA WAS TO ESTABLISH OVER A VARIETY OF DOMAINS THEORIES OF DIVISIBILITY THAT MIMICKED THAT OF ELEMENTARY NUMBER THEORY THE DEVELOPMENT OF THIS IDEA AS NEUMANN SHOWS WAS NOT UNILATERAL FOLLOWING ERNST KUMMER S INVEN TION OF IDEAL COMPLEX NUMBERS IN THE MID NINETEENTH CENTURY TWO RATHER DISTINCT APPROACHES EVOLVED ONE CHAMPIONED BY KARL WEIERSTRASS EGOR ZOLOTAREV KURT HENSEL KRULL AND OTHERS CENTERED ON THE IDEA OF EMBEDDING THE VARIOUS DOMAINS EITHER IN PRINCIPAL IDEAL DOMAINS OR IN DIRECT PRODUCTS OF PRINCIPLE IDEAL DOMAINS THE OTHER GREW OUT OF DEDEKIND S THEORY OF IDEALS AND ENGAGED A STRING OF MATH EMATICIANS FROM DAVID HILBERT THROUGH ADOLF HURWITZ AND FRANCIS MACAULAY TO EMMY NOETHER THUS WHILE THE DIRECT LINE TO NOETHER IS EXPLORED SO IS ANOTHER EQUALLY IMPORTANT INTERTWINING LINE THE TWO LINES TAKEN TOGETHER PLAYED A KEY ROLE IN THE DEVELOPMENT OF THE MODERN CONCEPTS OF COMMUTATIVE RING THEORY ANOTHER KEY NINETEENTH CENTURY GURE WHO STANDS LARGELY O OF THE DIRECT LINE TO NOETHER IS LEOPOLD KRONECKER IN HIS CHAPTER KRONECKER S FUNDAMENTAL THEOREM OF GENERAL ARITHMETIC HAROLD EDWARDS EXPLORES JUST WHAT KRONECKER MEANT BY THIS THEOREM WHICH IS A REPLACEMENT FOR THE FUNDAMENTAL THEOREM OF ALGEBRA THE QUESTION IS HOW TO COMPUTE WITH ROOTS OF A POLYNOMIAL EQUATION OR EQUIVALENTLY HOW TO FACTOR POLYNOMIALS WITH COE CIENTS IN AN ALGEBRAIC NUMBER ELD KRONECKER IN KEEPING WITH HIS CONSTRUCTIVIST APPROACH TO MATHEMATICS AD VOCATED NOT REGARDING THESE ROOTS AS REAL NUMBERS AND THEREFORE HAD TO DEVELOP A METHOD FOR WORKING WITH THEM HE SUCCEEDED IN HIS TASK AND SAW HIS APPROACH AS LYING AT THE HEART OF WHAT HE CALLED THE PRECIOUS GALOIS PRINCIPLE THIS LED KRONECKER TO HIS NEW FUNDAMENTAL THEOREM AND EDWARDS HERE NOT ONLY GIVES HIS OWN ACCOUNT OF IT BUT ALSO USES THIS ACCOUNT TO SHED LIGHT ON HIS OWN UNDERSTANDING OF SOME OF KRONECKER S MURKIER ARGUMENTS HISTORY OF MATHEMATICS 4