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Of Niels Henrik Abel Olav Arnfinn Laudal- Ragni Piene (Editors) The Legacy of Niels Henrik Abel The Abel Bicentennial, Oslo, 2002 Springer Editors: Olav Arnfinn Laudal Ragni Piene University of Oslo Department of Mathematics 0316 Oslo, Norway e-mail: [email protected] URL: http://folk.uio.no/arnfinnll , http://folk. uio.no/ragnip/ I ~- Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliotheklists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2000): Ol-XX, ll-XX, 14-XX, 16-XX, 32-XX, 34-XX, 37-XX, 51-XX ISBN 3-540-43826-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965. in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Please note: The software is protected by copyright. The publisher and the authors accept no legal responsibility for any damage caused by improper use of the instructions and programs contained in this book and the CD-ROM. Although the software has been tested. with extreme care, errors in the software cannot be excluded. Decompiling, disassembling, reverse engineering or in any way changing the program is expressly forbidden. For more details concerning the conditions of use and warranty we refer to the License Agreement on the CD-ROM (license.lXt). Springer-Verlag is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, regis'tered names) trademarks etc. in this publication does not imply, even in th e absence of a specific statement. that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Berlin Typesetting: Le-TeX Jelonek, Schmidt & Wielder GbR, Leipzig Printed on acid-free paper 46/3142db-5 43210 Contents The Legacy of Niels Henrik Abel .............. ......... .. .... King Harald V Opening address The Abel Bicentennial Conference University of Oslo, June 3, 2002 . 5 Arild Stubhaug The Life of Niels Henrik Abel . 7 Christian HouzeI The Work of Niels Henrik Abel . 21 PhiIlip Griffiths The Legacy of Abel in Algebraic Geometry ........ .. ........... .. 179 Daniel Lazard Solving Quintics by Radicals . .. .... .... ... .. ... ..... .. .... .. .. 207 Birgit Petri and Norbert Schappacher From Abel to Kronecker: Episodes from 19th Century Algebra ......... 227 Giinther Frei On the History of the Artin Reciprocity Law in Abelian Extensions of Algebraic Number Fields: How Artin was Led to his Reciprocity Law . 267 Aldo Brigaglia, Ciro Cmberto, Claudio Pedrini The Italian School of Algebraic Geometry and Abel's Legacy .. .... .. 295 Fabrizio Catanese From Abel's Heritage: Transcendental Objects in Algebraic Geometry and Their Algebraization ... .. .. 349 Steven L. Kleiman What is Abel's Theorem Anyway? . .. 395 Torsten Ekedahl On Abel's Hyperelliptic Curves ..... ......... ................ 441 VI Contents Mark Green, Phillip Griffiths Formal Deformation of Chow Groups . .. 467 Herbert Clemens An Analogue of Abel's Theorem ................................. 511 Tom Graber, Joe Harris, Barry Mazur, Jason Starr Arithmetic Questions Related to Rationally Connected Varieties ... ... 531 Yum-Tong Siu Hyperbolicity in Complex Geometry . 543 G.M.Henkin Abel-Radon Transform and Applications . .. ... 567 Simon Gindikin Abel Transform and Integral Geometry . .. 585 v. P. Palamodov Abel's Inverse Problem and Inverse Scattering .. .. 597 Jan-Erik Bjork Residues and i)-modules . .. 605 Mikael Passare, August Tsikh Algebraic Equations and Hypergeometric Series. .. 653 mlkan Hedenmalm Dirichlet Series and Functional Analysis. 673 Yuri I. Manin Real Multiplication and Noncommutative Geometry .... .......... .. 685 William FuIton On the Quantum Cohomology of Homogeneous Varieties . .. 729 Christian Kassel Quantum Principal Bundles up to Homotopy Equivalence . .. 737 Michel van den Bergh Non-commutative Crepant Resolutions .. ... .. .......... .... .. 749 Moira Chas, Dennis Sullivan Closed String Operators in Topology Leading to Lie Bialgebras and Higher String Algebra . .. 771 From Abel to Kronecker: Episodes from 19th Century Algebra Birgit Petri and Norbert Schappacher Recalling Niels Henrik Abel on the Resolution of Algebraic Equations 2 1853-1856: The General Form of Solvable and.Cyclic Equations 3 1858-1861: Elliptic Functions and the Quintic Equation 4 1861-192 1: Leopold Kronecke.r vs. Felix Klein 5 Glimpses Beyond References This paper is about Leopold Kronecker reading Niels Henrik Abel's results and ideas on the resolution of algebraic equations. When the young Kronecker began to work on algebraic equations, he went back and forth between Abel's works and his own ideas. And throughout his career he continued to position himself much nearer to Abel than to Galois. At the same time, his own creativity transformed Abel's results and questions into something more arithmetic and fairly different. For instance, already in his very first publication on algebraic equations [41], when unfolding Abet's problems on solvable equations, Kronecker essentially claimed both what is known today as the 'Theorem of Kronecker and Weber,' to the effect that every abelian extension of Q is cyclotornic, and its analogue for abelian extensions of QC.J=l), and even indicated further generalizations. Our article was written with two different kinds of readers in mind: interested mathematicians as well as historians of mathematics. The mathematician, espe­ cially the number theorist, may appreciate for example our attempt at partially reconstructing Kronecker's reasoning for the priine-degree-case of the 'Kronecker­ Weber-Theorem'. The historian will notice the newly used documents. For instance, we draw on, and publish here for the first time, a few letters from Kronecker to Dirichlet, of which Harold M. Edwards has made us aware and shared his personal transcription with us. We also make use of some of the unpublished, handwrit­ ten notes of Kronecker's lecture courses which are preserved in the library of the Strasbourg mathematical institute IRMA. Both the mathematical and the historical reader, however, will not fail to realize the peculiar position that the mathematical events discussed here occupy with respect to what can be considered (at least with hindsight) as the mainstream development 228 B. Petri and N. Schappacher of Galois Theory in the nineteenth century. Comparing for instance Kronecker's first paper in the subject [41J to Enrico Betti's big memoir [9], which had appeared only a year before, and which Kronecker knew, the two texts almost seem to belong to different mathematical cultures: Betti's treatise is justly regarded to be a milestone in the development of Galois Theory in that it treats permutations first - if in a way which is still quite far from our group theory, and actually quite hard to penetrate for mathematician and historian alike -, and their applications to the theory of algebraic equations in a separate part thereafter. Kronecker's Berlin Academy Note on the other hand barely sketches proofs, and takes a dramatic number theoretic turn at the end, which may well have been the very starting point of his work. The final Sect. 4 of our paper concentrates on elements of the history of Kro­ necker's 1861 sharpening of Abel's theorem on the non-resolubility of the gen­ eral quintic equation: Kronecker saw that such an equation does not admit one­ parameter resolvents. This was preceded by several proposals - due to Hermite, Kronecker, and Brioschi - to use elliptic functions in the resolution of the quin­ tic equation; we will describe them briefly in Sect. 3. On the other hand, Kro­ necker's theorem of 1861 gave rise to a lasting difference of opinions between Kronecker and Felix Klein about the limits of algebraic resolutions of the quin­ tic equation. It was Klein who supplied the first published proof of Kroneck­ er's result; but we will try to explain the difference of his point of view from Kronecker's. The present paper thus highlights a few seminal episodes from nineteenth century algebra, which are directly linked to the name of Abel, and which, even though partly forgotten, are part of our mathematical heritage. 1 Recalling Niels Henrik Abel on the Resolution of Algebraic Equations 1.1. Since early mathematics in several different cultures had some knowledge of solving what we consider today as special cases of quadratic polynomial equations, it is probably impossible to pin down a precise first occurence of the general for­ mula for the solutions of all quadratic equations x 2 + px + q = 0, i.e., of the formula p 1 K ~ X = -- + -(-1) y L1 (for K = 0, 1), K 2 2 where L1 = p2 - 4q, and where the square root is fixed in some way. 1.2. The resolution of cubic equations was largely accomplished by Tartaglia and Cardano around 1540. In today's notation, if x 3 + px + q = 0 (a case to which one is easily reduced by what was later to be called a Tschirnhausen transformation), From Abel to Kronecker 22l} and if we set .1 = -4 p3 - 27 q2, then, for a suitable choice of the two cubic roots, one obtains the solutions: (forK = 0,1,2), where Q is a primitive third root of 1. I 1.3. Skipping the case of degree four (in fact, we will only consider equa­ tions of prime degree further on), the next bit of general mathematical cul­ ture in the theory of algebraic equations is of course the impossibility of re­ solving the general quintic equation by radicals, which was proved com­ pletely for the first time by Abel - see [2].
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