Springer Monographs in Mathematics Władysław Narkiewicz The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate Springer Monographs in Mathematics Editors-in-chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series editors Sheldon Axler, San Francisco, USA Mark Braverman, Toronto, Canada Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto Pinto, Porto, Portugal Gabriella Pinzari, Napoli, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK Thisseriespublishesadvancedmonographsgivingwell-writtenpresentationsofthe “state-of-the-art”infieldsofmathematicalresearchthathaveacquiredthematurity neededforsuchatreatment.Theyaresufficientlyself-containedtobeaccessibleto morethanjusttheintimatespecialistsofthesubject,andsufficientlycomprehensive to remain valuable references for many years. Besides the current state of knowledgeinitsfield,anSMMvolumeshouldideallydescribeitsrelevancetoand interaction with neighbouring fields of mathematics, and give pointers to future directions of research. More information about this series at http://www.springer.com/series/3733 ł ł W adys aw Narkiewicz The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate 123 Władysław Narkiewicz University of Wrocław Wrocław,Poland ISSN 1439-7382 ISSN 2196-9922 (electronic) SpringerMonographs inMathematics ISBN978-3-030-03753-6 ISBN978-3-030-03754-3 (eBook) https://doi.org/10.1007/978-3-030-03754-3 LibraryofCongressControlNumber:2018960727 MathematicsSubjectClassification(2010): 11Rxx,11-03,01A60 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To the memory of my wife Preface Theaimofthisbookistogiveasurveyofresultsinthetheoryofalgebraicnumbers achieved in the first half of the twentieth century and may be viewed as a com- paniontomypreviousbookRationalNumberTheoryinthe20thCenturyinwhich the part of number theory dealing with rational numbers has been treated. It is an attempt to fulfil the wish of H. S. Vandiver expressed in 1960 in his paper [4185], and perhaps it might be helpful in preventing rediscoveries. Chapter 1 gives a concise presentation of the beginnings of the theory of algebraicnumbers.Onefindsherefirstadescriptionoftheworkonspecialcasesof algebraicintegersdonebyGauss,DirichletandEisenstein,followedbyKummer’s work on cyclotomic fields. Then the creation of the general theory by Kronecker and Dirichlet is treated, and the chapter concludes with a short description of the related work of other mathematicians, including Hermite, Minkowski, Frobenius and Stickelberger. InChap.2 onefindsapresentation oftheworkof Hilbert,who inhis report on algebraicnumberssummarizedthestateoftheirtheoryattheendofthenineteenth century, as well of Hensel, who created p-adic and p-adic numbers, which turned outtobeanindispensabletoolinfutureresearch.Inthelastpartofthechapterthe first steps towards creation of the class-field theory, characterizing Abelian exten- sions of algebraic number fields, are described. Chapter 3 covers the first twenty years of the twentieth century. In the first section we present its central subject, the use of analytic methods in the theory of algebraic numbers. This has been initiated by Landau, who established the Prime Ideal Theorem giving asymptotics for the number of prime ideal with bounded norms. The next big achievement was Hecke’s proof of the continuation of the Dedekind zeta-function to a meromorphic function on the plane, and the study of severalgeneralizationsofDirichletL-functionstonumberfields,developedbyhim and Landau. The second section presents the results dealing with the algebraic structure, and the last section is devoted to other results achieved in the beginning of the twentieth century. vii viii Preface The central themes of Chap. 4 are the creation of the modern ideal theory by EmmyNoether,andtheestablishmentoffundamentaltheoremsofclass-fieldtheory byTakagiandArtin.Alsovariousotherquestionswereconsideredatthattime,for example the first results in the additive theory of algebraic numbers obtained by Rademacher. In Chap. 5 we present first the progress in the study of the structure of number fields, the central subject being the existence of normal and normal integral bases, and then consider some additive questions, mainly on sums of squares. The next section concentrates on the simplification of the class-field theory by Hasse and Chevalley,andthefollowingsectionsconcerni.a.theclass-numberandclass-group of quadratic fields, the question of the existence of the Euclidean algorithm, the distribution of algebraic integers on the complex plane and infinite extensions of number fields. Chapter 6 covers The Forties, the main results being obtained by Brauer and Siegel. Inallchaptersonewillfindalsosomeselectedinformationaboutthesubsequent developments of the arising problems. IamverygratefultomyfriendsKálmanGyőryandAndrzejSchinzelforreading thedraftofthebookandprovidingseveralcommentsandsuggestions.Ithankalso the referees of the book for several important hints. I am very grateful to the Springer stafffor helpful cooperation in preparing the publication. My specials thanks go to Ms Elena Griniari and Ms Angela Schulze-Thomin. Wrocław, Poland Władysław Narkiewicz Contents 1 The Birth of Algebraic Number Theory . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Euler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 First Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.1 Eisenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.2 Kummer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Establishing the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.3.1 Kronecker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.3.2 Geometrical Approach: Hermite and Minkowski . . . . . . . . 36 1.3.3 Dedekind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.3.4 Frobenius and Stickelberger . . . . . . . . . . . . . . . . . . . . . . . 53 1.4 Other Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.5 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2 The Turn of the Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.1 David Hilbert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.1.1 First Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.1.2 Zahlbericht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.1.3 After the Zahlbericht . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.2 Kurt Hensel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.2.1 Field Index and Monogenic Fields . . . . . . . . . . . . . . . . . . 73 2.2.2 Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.2.3 p-Adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.3 The Beginnings of Class-Field Theory. . . . . . . . . . . . . . . . . . . . . 85 2.3.1 Kronecker’s Jugendtraum. . . . . . . . . . . . . . . . . . . . . . . . . 85 ix x Contents 2.3.2 Heinrich Weber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.3.3 Hilbert’s Class-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3 First Years of the Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1 Analytic Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1.1 Edmund Landau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1.2 Erich Hecke and the New L-Functions . . . . . . . . . . . . . . . 103 3.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.2.1 Steinitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.2.2 Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.2.3 Discriminants and Integral Bases . . . . . . . . . . . . . . . . . . . 120 3.2.4 Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.2.5 Splitting Primes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.2.6 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.3 Class-Number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.3.1 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.3.2 Cyclotomic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.4 Other Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.5 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4 The Twenties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.1.1 Ideal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.1.2 Integral Bases, Discriminants, Factorizations . . . . . . . . . . . 144 4.1.3 Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2 Analytical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.2.1 Quadratic Reciprocity Law. . . . . . . . . . . . . . . . . . . . . . . . 153 4.2.2 Sums of Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.2.3 Sums of Primes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.2.4 Piltz Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.2.5 Values of Zeta-Functions . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.3 Class-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.3.1 Takagi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.3.2 Artin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.3.3 Hasse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.4 Class-Number and Class-Group. . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.4.1 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.4.2 Other Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.5 Other Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.5.1 Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.5.2 Algebraic Numbers in the Plane . . . . . . . . . . . . . . . . . . . . 182 4.5.3 Infinite Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.5.4 Varia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.5.5 Books. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186