DOCSLIB.ORG

Gauss's Hidden Menagerie: From

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Gauss's Hidden Menagerie: From Gauss’s Hidden Menagerie: From Cyclotomy to Supercharacters Stephan Ramon Garcia, Trevor Hyde, and Bob Lutz t the age of eighteen, Gauss established the constructibility of the 17-gon, a result that had eluded mathematicians for two millennia. At the heart of his Aargument was a keen study of cer- tain sums of complex exponentials, known now as Gaussian periods. These sums play starring roles in applications both classical and modern, including Kummer’s development of arithmetic in the cyclotomic integers [28] and the optimized AKS primality test of H. W. Lenstra and C. Pomer- ance [1, 32]. In a poetic twist, this recent application of Gaussian periods realizes “Gauss’s dream” of an efficient algorithm for distinguishing prime (a) n = 29 · 109 · 113, ! = 8862, c = 113 numbers from composites [24]. We seek here to study Gaussian periods from a graphical perspective. It turns out that these clas- sical objects, when viewed appropriately, exhibit a dazzling and eclectic host of visual qualities. Some images contain discretized versions of familiar shapes, while others resemble natural phenomena. Many can be colorized to isolate certain features; for details, see “Cyclic Supercharacters.” Historical Context The problem of constructing a regular polygon with compass and straight-edge dates back to (b) n = 37 · 97 · 113, ! = 5507, c = 113 ancient times. Descartes and others knew that with only these tools on hand, the motivated geometer Figure 1. Eye and jewel—images of cyclic supercharacters correspond to sets of Gaussian could draw, in principle, any segment whose periods. For notation and terminology, see length could be written as a finite composition “Cyclic Supercharacters.” Stephan Ramon Garcia is associate professor of mathe- matics at Pomona College. His email address is Stephan. [email protected]. of sums, products, and square roots of rational Trevor Hyde is a graduate student at the University of Michi- gan. His email address is [email protected]. numbers [18]. Gauss’s construction of the 17-gon Bob Lutz is a graduate student at the University of Michi- relied on showing that gan. His email address is [email protected]. All article 2π p q p figures are courtesy of Bob Lutz. 16 cos = −1 + 17 + 34 − 2 17 Partially supported by National Science Foundation Grant 17 r DMS-1265973. p q p q p + + − − − + DOI: http://dx.doi.org/10.1090/noti1269 2 17 3 17 34 2 17 2 34 2 17 878 Notices of the AMS Volume 62, Number 8 (a) n = 3 · 5 · 17 · 29 · 37], ! = 184747, c = 3 · 17 (b) n = 13 · 127 · 199, ! = 6077, c = 13 Figure 2. Disco ball and loudspeaker—images of cyclic supercharacters correspond to sets of Gaussian periods. For notation and terminology, see “Cyclic Supercharacters.” was such a length. After reducing the constructibil- wrote that “[c]yclotomy is to be regarded … as 2π the natural and inherent centre and core of the ity of the n-gon to drawing the length cos n , his result followed easily. So proud was Gauss of this arithmetic of the future” [39]. Two of Kummer’s discovery that he wrote about it throughout his most significant achievements depended critically career, purportedly requesting a 17-gon in place on his study of Gaussian periods: Gauss’s work of his epitaph.1 While this appeal went unfulfilled, laid the foundation for the proof of Fermat’s Last sculptor Fritz Schaper did include a 17-pointed star Theorem in the case of regular primes and later at the base of a monument to Gauss in Brunswick, for Kummer’s celebrated reciprocity law. where the latter was born [31]. This success inspired Kummer to generalize Gauss went on to demonstrate that a regular Gaussian periods in [30] to the case of composite n-gon is constructible if Euler’s totient '(n) is moduli. Essential to his work was a study of d a power of 2. He stopped short of proving that the polynomial x − 1 by his former student, these are the only cases of constructibility; this L. Kronecker, whom Kummer continued to mentor remained unsettled until J. Petersen completed a for the better part of both men’s careers [27]. Just as largely neglected argument of P. Wantzel nearly Gaussian periods for prime moduli had given rise three quarters of a century later [33]. Nonetheless, to various families of difference sets [7], Kummer’s the chapter containing Gauss’s proof has persisted composite cyclotomy has been used to explain deservedly as perhaps the most well-known section certain difference sets arising in finite projective of his Disquisitiones Arithmeticae. Without the geometry [14]. Shortly after Kummer’s publication, language of abstract algebra, Gauss initiated the L. Fuchs presented a result in [23] concerning the study of cyclotomy, literally “circle cutting,” from vanishing of Kummer’s periods that has appeared an algebraic point of view. in several applications by K. Mahler [34], [35]. A The main ingredient in Gauss’s argument is modern treatment of Fuchs’s result and a further an exponential sum known as a Gaussian period. generalization of Gaussian periods can be found Denoting the cardinality of a set S by jSj, if p is in [21]. an odd prime number and ! has order d in the For a positive integer n and positive divisor unit group (Z=pZ)×, then the d-nomial Gaussian d of '(n), Kummer “defined” a d-nomial period periods modulo p are the complex numbers modulo n to be the sum d−1 j ! d−1 j ! X ! y X ! y (1) e ; e ; n p j=0 j=0 × where y belongs to Z=pZ and e(θ) denotes where ! has order d in the unit group (Z=nZ) and exp(2πiθ) for real θ. Following its appearance in y ranges over Z=nZ. Unlike the case of prime moduli, Disquisitiones, Gauss’s cyclotomy drew the atten- however, there is no guarantee that a generator ! tion of other mathematicians who saw its potential of order d will exist or that a subgroup of order × use in their own work. In 1879, J. J. Sylvester d will be unique. For example, consider (Z=8Z) , which contains no element of order 4, as well as 1H. Weber makes a footnote of this anecdote in [43, p. 362], three distinct subgroups of order 2. A similar lack but omits it curiously from later editions. of specificity pervaded some of Kummer’s other September 2015 Notices of the AMS 879 definitions, including his introduction of ideal We are now in a position to clarify the captions prime factors, used to prove a weak form of prime and colorizations of the numerous figures. Unless factorization for cyclotomic integers. According specified otherwise, the image appearing in each to H. M. Edwards, instead of revealing deficiencies figure is the image in C of the cyclic supercharacter × in Kummer’s work, these examples suggest “the σh!i : Z=nZ ! C, where ! belongs to (Z=nZ) , and mathematical culture…as Kummer saw it” [19]. h!i = h!i1 denotes the orbit of 1 under the action Fortunately, the ambiguity in Kummer’s defi- of the subgroup generated by !. Conveniently, nition is easily resolved. For n as above and an the image of any cyclic supercharacter is a scaled × element ! of (Z=nZ) , we define the Gaussian subset of the image of one having the form σh!i [17, periods generated by ! modulo n to be the sums Proposition 2.2], so a restriction of our attention in (1), where d is the order of ! and y ranges to orbits of 1 is natural. Moreover, the image of over Z=nZ, as before. These periods are closely σh!i is the set of Gaussian periods generated by related to Gauss sums, another type of exponential ! modulo n, bringing classical relevance to these sum [9]. figures. To colorize each image, we fix a proper divisor Cyclic Supercharacters c of n and assign a color to each of the layers, In 2008, P. Diaconis and I. M. Isaacs introduced σ (y) j y ≡ j (mod c) ; the theory of supercharacters axiomatically [15], h!i1 building upon seminal work of C. André on the rep- for j = 0; 1; : : : ; c − 1. Different choices of c result resentation theory of unipotent matrix groups [3], in different “layerings.” For many images, certain [4]. Supercharacter techniques have been used to values of c yield colorizations that separate distinct study the Hopf algebra of symmetric functions graphical components. of noncommuting variables [2], random walks on upper triangular matrices [5], combinatorial prop- erties of Schur rings [16], [40], [41], and Ramanujan sums [22]. To make an important definition, we divert briefly to the character theory of finite groups. Let G be a finite group with identity 0, K a partition of G, and X a partition of the set of irreducible characters of G. The ordered pair (X; K) is called a supercharacter theory for G if f0g 2 K, jXj = jKj, and for each X 2 X, the function X σX = χ(0)χ χ2X is constant on each K 2 K. The functions σX : G ! C are called supercharacters, and the elements of K are called superclasses. (a) n = 13 · 127 · 199, X = ! = 9247, c = 127 Since Z=nZ is abelian, its irreducible characters are group homomorphisms Z=nZ ! C×. Namely, for each x in Z=nZ, there is an irreducible character xy χx of Z=nZ given by χx(y) = e( n ). For a subgroup × Ð of (Z=nZ) , let K denote the partition of Z=nZ arising from the action a · x = ax of Ð . The action a · χx = χa−1x of Ð on the irreducible characters of Z=nZ yields a compatible partition X making (X; K) a supercharacter theory on Z=nZ.
Load more

Recommended publications
  • La Controverse De 1874 Entre Camille Jordan Et Leopold Kronecker
    La controverse de 1874 entre Camille Jordan et Leopold Kronecker. * Frédéric Brechenmacher ( ). Résumé. Une vive querelle oppose en 1874 Camille Jordan et Leopold Kronecker sur l’organisation de la théorie des formes bilinéaires, considérée comme permettant un traitement « général » et « homogène » de nombreuses questions développées dans des cadres théoriques variés au XIXe siècle et dont le problème principal est reconnu comme susceptible d’être résolu par deux théorèmes énoncés indépendamment par Jordan et Weierstrass. Cette controverse, suscitée par la rencontre de deux théorèmes que nous considèrerions aujourd’hui équivalents, nous permettra de questionner l’identité algébrique de pratiques polynomiales de manipulations de « formes » mises en œuvre sur une période antérieure aux approches structurelles de l’algèbre linéaire qui donneront à ces pratiques l’identité de méthodes de caractérisation des classes de similitudes de matrices. Nous montrerons que les pratiques de réductions canoniques et de calculs d’invariants opposées par Jordan et Kronecker manifestent des identités multiples indissociables d’un contexte social daté et qui dévoilent des savoirs tacites, des modes de pensées locaux mais aussi, au travers de regards portés sur une histoire à long terme impliquant des travaux d’auteurs comme Lagrange, Laplace, Cauchy ou Hermite, deux philosophies internes sur la signification de la généralité indissociables d’idéaux disciplinaires opposant algèbre et arithmétique. En questionnant les identités culturelles de telles pratiques cet article vise à enrichir l’histoire de l’algèbre linéaire, souvent abordée dans le cadre de problématiques liées à l’émergence de structures et par l’intermédiaire de l’histoire d’une théorie, d’une notion ou d’un mode de raisonnement.
    [Show full text]
  • Kummer, Regular Primes, and Fermat's Last Theorem
    e H a r v a e Harvard College r d C o l l e Mathematics Review g e M a t h e m Vol. 2, No. 2 Fall 2008 a t i c s In this issue: R e v i ALLAN M. FELDMAN and ROBERTO SERRANO e w , Arrow’s Impossibility eorem: Two Simple Single-Profile Versions V o l . 2 YUFEI ZHAO , N Young Tableaux and the Representations of the o . Symmetric Group 2 KEITH CONRAD e Congruent Number Problem A Student Publication of Harvard College Website. Further information about The HCMR can be Sponsorship. Sponsoring The HCMR supports the un- found online at the journal’s website, dergraduate mathematics community and provides valuable high-level education to undergraduates in the field. Sponsors http://www.thehcmr.org/ (1) will be listed in the print edition of The HCMR and on a spe- cial page on the The HCMR’s website, (1). Sponsorship is available at the following levels: Instructions for Authors. All submissions should in- clude the name(s) of the author(s), institutional affiliations (if Sponsor $0 - $99 any), and both postal and e-mail addresses at which the cor- Fellow $100 - $249 responding author may be reached. General questions should Friend $250 - $499 be addressed to Editors-In-Chief Zachary Abel and Ernest E. Contributor $500 - $1,999 Fontes at [email protected]. Donor $2,000 - $4,999 Patron $5,000 - $9,999 Articles. The Harvard College Mathematics Review invites Benefactor $10,000 + the submission of quality expository articles from undergrad- uate students.
    [Show full text]
  • Methodology and Metaphysics in the Development of Dedekind's Theory
    Methodology and metaphysics in the development of Dedekind’s theory of ideals Jeremy Avigad 1 Introduction Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical fac- tions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the oppos- ite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place.
    [Show full text]
  • The Sextic Period Polynomial Andrew J
    BULL. AUSTRAL. MATH. SOC. 11R20, 11L99 VOL. 49 (1994) [293-304] THE SEXTIC PERIOD POLYNOMIAL ANDREW J. LAZARUS In this paper we show that the method of calculating the Gaussian period poly- nomial which originated with Gauss can be replaced by a more general method based on formulas for Lagrange resolvants. The period polynomial of cyclic sextic fields of arbitrary conductor is determined by way of example. 1. INTRODUCTION Suppose p = ef + 1 is prime. Define the e cyclotomic classes where g is any primitive root modulo p. The Gaussian periods rjj are defined by (1.1) W The principal class Co contains the e-th power residues and the other classes are its cosets. The rjj are Galois conjugates and the period polynomial \Pe(X) is their com- mon minimal polynomial over Q. Gauss introduced the cyclotomic numbers (A,fc) determined, for a given g, by (h, k) = #{v e (Z/JIZ)* : v e Ch, v + 1 G Ck}. It follows that e-l (1.2) »?o»7/, = X; (&,%*+MM) Jfe=0 where S is Kronecker's delta and I — 0 or e/2 according as / is even or odd. The coefficients of ^s(X) in terms of p and the coefficients of the quadratic form 4p = A2 + 27B2 were determined by Gauss in Disquisitiones Arithmetical enough rela- tions exist to determine all (h,k) in terms of p, A, and B. The period polynomial's Received 10th May, 1993 Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/94 SA2.00+0.00. 293 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • The History of the Formulation of Ideal Theory
    The History of the Formulation of Ideal Theory Reeve Garrett November 28, 2017 1 Using complex numbers to solve Diophantine equations From the time of Diophantus (3rd century AD) to the present, the topic of Diophantine equations (that is, polynomial equations in 2 or more variables in which only integer solutions are sought after and studied) has been considered enormously important to the progress of mathematics. In fact, in the year 1900, David Hilbert designated the construction of an algorithm to determine the existence of integer solutions to a general Diophantine equation as one of his \Millenium Problems"; in 1970, the combined work (spanning 21 years) of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson showed that no such algorithm exists. One such equation that proved to be of interest to mathematicians for centuries was the \Bachet equa- tion": x2 +k = y3, named after the 17th century mathematician who studied it. The general solution (for all values of k) eluded mathematicians until 1968, when Alan Baker presented the framework for constructing a general solution. However, before this full solution, Euler made some headway with some specific examples in the 18th century, specifically by the utilization of complex numbers. Example 1.1 Consider the equation x2 + 2 = y3. (5; 3) and (−5; 3) are easy to find solutions, but it's 2 not obviousp whetherp or not there are others or whatp they might be. Euler realized by factoring x + 2 as (x + 2i)(x − 2i) and then using the facts that Z[ 2i] is a UFD andp the factorsp given are relatively prime that the solutions above are the only solutions,p namelyp because (x + 2i)(x − 2i) being a cube forces each of these factors to be a cube (i.e.
    [Show full text]
  • Computing Isomorphisms and Embeddings of Finite
    Computing isomorphisms and embeddings of finite fields Ludovic Brieulle, Luca De Feo, Javad Doliskani, Jean-Pierre Flori and Eric´ Schost May 4, 2017 Abstract Let Fq be a finite field. Given two irreducible polynomials f; g over Fq, with deg f dividing deg g, the finite field embedding problem asks to compute an explicit descrip- tion of a field embedding of Fq[X]=f(X) into Fq[Y ]=g(Y ). When deg f = deg g, this is also known as the isomorphism problem. This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better. We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software. Contents 1 Introduction2 2 Preliminaries4 2.1 Fundamental algorithms and complexity . .4 2.2 The Embedding Description problem . 11 arXiv:1705.01221v1 [cs.SC] 3 May 2017 3 Kummer-type algorithms 12 3.1 Allombert's algorithm . 13 3.2 The Artin{Schreier case . 18 3.3 High-degree prime powers . 20 4 Rains' algorithm 21 4.1 Uniquely defined orbits from Gaussian periods . 22 4.2 Rains' cyclotomic algorithm . 23 5 Elliptic Rains' algorithm 24 5.1 Uniquely defined orbits from elliptic periods .
    [Show full text]
  • L-Functions and Non-Abelian Class Field Theory, from Artin to Langlands
    L-functions and non-abelian class field theory, from Artin to Langlands James W. Cogdell∗ Introduction Emil Artin spent the first 15 years of his career in Hamburg. Andr´eWeil charac- terized this period of Artin's career as a \love affair with the zeta function" [77]. Claude Chevalley, in his obituary of Artin [14], pointed out that Artin's use of zeta functions was to discover exact algebraic facts as opposed to estimates or approxi- mate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin L-functions as a tool for finding a non- abelian class field theory, expressed as the desire to extend results from relative abelian extensions to general extensions of number fields. Artin introduced his L-functions attached to characters of the Galois group in 1923 in hopes of developing a non-abelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law - the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a non-abelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 \Artin stated that `My own belief is that we know it already, though no one will believe me { that whatever can be said about non-Abelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields { and there are sufficiently many Abelian cases.' The critical thing is learning how to pass from a prime in an intermediate field to a prime in a large field.
    [Show full text]
  • Dedekind's 1871 Version of the Theory of Ideals∗
    Dedekind's 1871 version of the theory of ideals¤ Translated by Jeremy Avigad March 19, 2004 Translator's introduction By the middle of the nineteenth century, it had become clear to mathe- maticians that the study of ¯nite ¯eld extensions of the rational numbers is indispensable to number theory, even if one's ultimate goal is to understand properties of diophantine expressions and equations in the ordinary integers. It can happen, however, that the \integers" in such extensions fail to satisfy unique factorization, a property that is central to reasoning about the or- dinary integers. In 1844, Ernst Kummer observed that unique factorization fails for the cyclotomic integers with exponent 23, i.e. the ring Z[³] of inte- gers of the ¯eld Q(³), where ³ is a primitive twenty-third root of unity. In 1847, he published his theory of \ideal divisors" for cyclotomic integers with prime exponent. This was to remedy the situation by introducing, for each such ring of integers, an enlarged domain of divisors, and showing that each integer factors uniquely as a product of these. He did not actually construct these integers, but, rather, showed how one could characterize their behav- ior qua divisibility in terms of ordinary operations on the associated ring of integers. Richard Dedekind and Leopold Kronecker later took up the task of ex- tending the theory to the integers in arbitrary ¯nite extensions of the ratio- nals. Despite their common influences and goals, however, the theories they ¤Work on this translation has been supported by a New Directions fellowship from the Andrew W.
    [Show full text]
  • Sir Andrew Wiles Awarded Abel Prize
    Sir Andrew J. Wiles Awarded Abel Prize Elaine Kehoe with The Norwegian Academy of Science and Letters official Press Release ©Abelprisen/DNVA/Calle Huth. Courtesy of the Abel Prize Photo Archive. ©Alain Goriely, University of Oxford. Courtesy the Abel Prize Photo Archive. Sir Andrew Wiles received the 2016 Abel Prize at the Oslo award ceremony on May 24. The Norwegian Academy of Science and Letters has carries a cash award of 6,000,000 Norwegian krone (ap- awarded the 2016 Abel Prize to Sir Andrew J. Wiles of the proximately US$700,000). University of Oxford “for his stunning proof of Fermat’s Citation Last Theorem by way of the modularity conjecture for Number theory, an old and beautiful branch of mathemat- semistable elliptic curves, opening a new era in number ics, is concerned with the study of arithmetic properties of theory.” The Abel Prize is awarded by the Norwegian Acad- the integers. In its modern form the subject is fundamen- tally connected to complex analysis, algebraic geometry, emy of Science and Letters. It recognizes contributions of and representation theory. Number theoretic results play extraordinary depth and influence to the mathematical an important role in our everyday lives through encryption sciences and has been awarded annually since 2003. It algorithms for communications, financial transactions, For permission to reprint this article, please contact: and digital security. [email protected]. Fermat’s Last Theorem, first formulated by Pierre de DOI: http://dx.doi.org/10.1090/noti1386 Fermat in the seventeenth century, is the assertion that 608 NOTICES OF THE AMS VOLUME 63, NUMBER 6 the equation xn+yn=zn has no solutions in positive integers tophe Breuil, Brian Conrad, Fred Diamond, and Richard for n>2.
    [Show full text]
  • A History of Stickelberger's Theorem
    A History of Stickelberger’s Theorem A Senior Honors Thesis Presented in Partial Fulfillment of the Requirements for graduation with research distinction in Mathematics in the undergraduate colleges of The Ohio State University by Robert Denomme The Ohio State University June 8, 2009 Project Advisor: Professor Warren Sinnott, Department of Mathematics 1 Contents Introduction 2 Acknowledgements 4 1. Gauss’s Cyclotomy and Quadratic Reciprocity 4 1.1. Solution of the General Equation 4 1.2. Proof of Quadratic Reciprocity 8 2. Jacobi’s Congruence and Cubic Reciprocity 11 2.1. Jacobi Sums 11 2.2. Proof of Cubic Reciprocity 16 3. Kummer’s Unique Factorization and Eisenstein Reciprocity 19 3.1. Ideal Numbers 19 3.2. Proof of Eisenstein Reciprocity 24 4. Stickelberger’s Theorem on Ideal Class Annihilators 28 4.1. Stickelberger’s Theorem 28 5. Iwasawa’s Theory and The Brumer-Stark Conjecture 39 5.1. The Stickelberger Ideal 39 5.2. Catalan’s Conjecture 40 5.3. Brumer-Stark Conjecture 41 6. Conclusions 42 References 42 2 Introduction The late Professor Arnold Ross was well known for his challenge to young students, “Think deeply of simple things.” This attitude applies to no story better than the one on which we are about to embark. This is the century long story of the generalizations of a single idea which first occurred to the 19 year old prodigy, Gauss, and which he was able to write down in no less than 4 pages. The questions that the young genius raised by offering the idea in those 4 pages, however, would torment the greatest minds in all the of the 19th century.
    [Show full text]
  • A Letter of Hermann Amandus Schwarz on Isoperimetric Problems
    Research Collection Journal Article A Letter of Hermann Amandus Schwarz on Isoperimetric Problems Author(s): Stammbach, Urs Publication Date: 2012-03 Permanent Link: https://doi.org/10.3929/ethz-b-000049843 Originally published in: The Mathematical Intelligencer 34(1), http://doi.org/10.1007/s00283-011-9267-7 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Years Ago David E. Rowe, Editor n 1995 (or thereabouts), Eli Maor, then at Loyola A Letter of Hermann University, discovered a little booklet at a used book fair IIin Chicago containing a reprint of a paper published by Hermann Amandus Schwarz in 1884 [Sch1]. Interestingly, it Amandus Schwarz also contained an original letter by Schwarz written in the old German script. The letter, dated January 28, 1884, began with the words: Hochgeehrter Herr Director! Maor could decipher on Isoperimetric enough of the text to realize that it dealt with isoperimetric problems, which further awakened his interest. With the help Problems of Reny Montandon and Herbert Hunziker, he contacted Gu¨nther Frei, who was able to read and transcribe the old URS STAMMBACH German script. Thanks to his detailed knowledge of the history of mathematics of the 19th century, Frei was also able to provide a number of pertinent remarks. Unfortunately, before completing this task, Gu¨nther Frei fell gravely ill. The text of the letter together with his remarks was then given to Years Ago features essays by historians and the present author, who agreed to continue the work.
    [Show full text]
  • Kummer, Regular Primes, and Fermat's Last Theorem 1
    KUMMER, REGULAR PRIMES, AND FERMAT'S LAST THEOREM ILA VARMA CALIFORNIA INSTITUTE OF TECHNOLOGY Abstract. This paper rephrases Kummer's proof of many cases of Fermat's last theorem in contemporary notation that was in fact derived from his work. Additionally, it develops a reformulation of the proof using class field theory from a modern perspective in a manner similar to the tactics used for the complete proof, and describes how Kummer's proof strategy can generalize to solve the theorem for a broader set of primes. 1. Introduction Ernst Kummer was a 19th century mathematician who came across Fermat's last theorem in attempts to generalize the law of quadratic reciprocity and study higher reciprocity laws. While he described those as \the principal subject and the pinnacle of contemporary number theory," he considered Fermat's last theorem a \curiosity of number theory rather than a major item" [1]. A priori, this was not an unreasonable opinion of a problem that could be understood by a 12-year-old. We state this mere curiosity below. Theorem. For any integer n > 2, the equation xn + yn = zn has no non-trivial solutions in the integers, i.e. if x; y; z 2 Z satisfy this equation, then xyz = 0. Despite his disinterest, Kummer made the first substantial step in proving a part of Fermat's last theorem for many cases. This came only a few weeks after Gabriel Lam´eincorrectly announced that he had found a complete proof [1]. Lam´edid make the breakthrough in attempting to decompose xn + yn into linear factors by introducing the complex numbers satisfying ζn = 1, known today as roots of unity.
    [Show full text]