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Computability in Principle and in Practice in Algebraic Number Theory: Hensel to Zassenhaus

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COMPUTABILITY IN PRINCIPLE AND IN PRACTICE IN ALGEBRAIC NUMBER THEORY: HENSEL TO ZASSENHAUS by Steven Andrew Kieffer M.Sc., Carnegie Mellon University, 2007 B.Sc., State University of New York at Buffalo, 2003 a Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science (Mathematics) in the Department of Mathematics Faculty of Science c Steven Andrew Kieffer 2012 SIMON FRASER UNIVERSITY Spring 2012 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for \Fair Dealing." Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Steven Andrew Kieffer Degree: Master of Science (Mathematics) Title of Thesis: Computability in principle and in practice in algebraic num- ber theory: Hensel to Zassenhaus Examining Committee: Dr. Jason Bell Chair Dr. Michael Monagan Senior Supervisor Professor Dr. Nils Bruin Supervisor Professor Dr. Tom Archibald Supervisor Professor Dr. J. Lennart Berggren Internal Examiner Professor Date Approved: 16 April 2012 ii Abstract In the early years of algebraic number theory, different mathematicians built the theory in terms of different objects, and according to different rules, some seeking always to demon- strate that the objects were computable in principle. Later, prominently in the era in which electronic computers were becoming available for academic research, efforts were initiated by some to compute the objects of the theory in practice. By examining writings, research, and correspondence of mathematicians spanning these early and late computational peri- ods, we seek to demonstrate ways in which ideas from the old tradition influenced the new. Among the connections we seek are personal influence on problem selection, and borrowing of computational methods. In particular, we examine such links among the works of Kurt Hensel, Helmut Hasse, Olga Taussky, and Hans Zassenhaus. iii For Mom and Dad iv \Wie man sich in der Musik nach der in heroischen und d¨amonischen Werken und in k¨uhnstenPhantasien schwelgenden romantischen und nachromantischen Epoche heute bei aller Freude an diesem Schaffen doch auch wieder st¨arker auf den Urquell reiner und schlichter Musikalit¨atder alten Meister besinnt, so scheint mir auch in der Zahlentheorie, die ja wie kaum eine andere mathematische Disziplin von dem Gesetz der Harmonie beherrscht wird, eine R¨uckbesinnung auf das geboten, was den großen Meistern, die sie begr¨undethaben, als ihr wahres Gesicht vorgeschwebt hat." | Helmut Hasse, 1945 v Acknowledgments I am grateful to my advisors Michael Monagan and Nils Bruin for sharing their expertise and enjoyment of computer algebra and number theory, for their careful comments and revisions, for the assignment to study Hensel's lemma, which has proved a more interesting topic than I ever expected, and for the gift of time in which to think about mathematics. I also thank my parents and friends for their support. Len Berggren generously gave his time to participate on the examining committee, and he and others in the department helped to make the history seminars interesting. Ideas from my former advisor Jeremy Avigad were surely at the back of my mind as I searched for interesting questions to ask about the methodology of mathematics. And finally, I would like to give very special thanks to Tom Archibald: for long discussions of historiographic questions, for a great deal of thoughtful feedback, for running the history seminars, and for his generous guidance and continual encouragement in the study and telling of history. vi Contents Approval ii Abstract iii Dedication iv Quotation v Acknowledgments vi Contents vii List of Tables xi List of Figures xii Preface xiii 1 Introduction 1 2 The Integral Basis 20 2.1 Kronecker's Grundz¨uge, 1882 . 22 2.2 Dedekind's Supplement XI, 1894 . 27 2.3 Hilbert's Zahlbericht, 1897 . 33 2.4 Hensel's Theorie der algebraischen Zahlen, 1908 . 36 2.4.1 TAZ Chapters 1 and 2 . 38 2.4.2 TAZ Chapters 3 and 4 . 40 2.4.3 TAZ Chapter 5 and the integral basis . 54 vii 2.4.4 Hensel's \subroutines" . 57 2.4.5 Algorithms and efficiency in Hensel . 59 3 Turn of the Century through World War II 65 3.1 Hensel and Hilbert . 66 3.2 Existence and construction of class fields . 75 3.3 Reciprocity laws . 86 3.3.1 Artin's law, explicit laws . 86 3.3.2 Hasse . 87 3.3.3 Artin . 89 3.3.4 Explicit reciprocity laws . 89 3.3.5 The Klassenk¨orperbericht . 92 3.4 Political unrest . 95 3.5 Olga Taussky . 97 3.6 Hans Zassenhaus . 103 4 Setting the stage in the 1940s 106 4.1 A photo-electric number sieve . 108 4.2 D.H. Lehmer's Guide to Tables, 1941 . 110 4.3 Weyl 1940 . 115 4.4 Hasse's view . 120 4.4.1 Hasse's Zahlentheorie . 121 4.4.2 Hasse's Uber¨ die Klassenzahl abelscher Zahlk¨orper . 123 5 Taussky-Hasse correspondence in the 1950s 139 5.1 Reestablishing contact . 143 5.2 Number theory on the computer in the early 1950s . 146 5.3 A survey talk, 1953 . 167 5.3.1 Rational methods . 170 5.3.2 Routine methods . 171 5.3.3 On generalization in number theory . 173 5.4 Letters from the 1960s and 70s . 174 5.5 Coda . 177 viii 6 Zassenhaus's work in the 1960s 178 6.1 Algebraic numbers in the computer { 1959 at Caltech . 183 6.2 Integral bases etc. in the mid 1960s . 190 6.2.1 Number theoretic experiments in education . 190 6.2.2 The ORDMAX algorithm { 1965 . 193 6.2.3 Berwick's algorithm . 195 6.2.4 Zassenhaus's algorithm . 197 6.2.5 \Round 2" . 200 6.3 On a problem of Hasse . 202 p 6.3.1 Background on the class field construction over Q( −47) . 204 6.3.2 Computing by machine and by hand . 209 6.3.3 The second class field construction . 210 6.3.4 Setting problems for Zassenhaus and Liang . 211 6.3.5 Solutions found . 213 6.3.6 The p-adic algorithm . 222 6.4 On Hensel factorization . 227 6.4.1 Zassenhaus's algorithm . 228 7 Epilogue 233 Appendix A Hensel's theory as generalization of Kummer's 244 A.1 Preliminaries . 247 A.2 Hensel's prime divisors . 260 A.3 Kummer's prime divisors . 263 A.4 Conclusions . 270 p Appendix B Hasse's first paper on Q( −47) 273 B.1 The Cases d = −23 and d = −31 . 274 B.1.1 Ascent to Generation of N 3=Ω3 . 275 B.1.2 Descent to Generation of K=Q . 277 B.2 The Case d = −47 . 279 B.2.1 Ascent to Generation of N 5=Ω5 . 279 B.2.2 Descent to Generation of K=Q . 285 ix Appendix C The Legendre-Germain computation 288 x List of Tables 1.1 Abbreviations for certain important works. 19 3.1 Early Jahresbericht data. 69 5.1 Data pertaining to the Legendre-Sophie-Germain criterion for ` = 79, follow- ing Hasse and Taussky, using primitive root 3 mod 79. 158 6.1 Years of Zassenhaus's earliest published contributions to his computational algebraic number theory program, by subject. 180 A.1 Correspondence for an example of Kummer's chemistry analogy. 272 xi List of Figures 3.1 Movements of selected mathematicians among selected German universities, 1841 to 1966. 67 3.2 Number of works cited per year in bibliography of Hilbert's Zahlbericht from 1825 forward, omitting three earlier references, from 1796, 1798, and 1801. 70 3.3 List of authors cited more than once in Hilbert's Zahlbericht, together with number of citations. 70 3.4 Number of papers published by Kurt Hensel each year, from 1884 to 1937. Source: (Hasse 1950b). 73 4.1 Number of papers published by Helmut Hasse each year, from 1923 to 1979. Source: (Hasse 1975) . 120 6.1 Number of papers published by Hans Zassenhaus each year, from 1934 to 1991. Source: (Pohst 1994) . 183 6.2 Flow chart for Zassenhaus's ORDMAX algorithm. 198 6.3 Field lattice for Hasse's problem. 207 7.1 Number of works cited per year in bibliography of Zimmer's survey (Zimmer 1972). 238 A.1 Field lattice for Henselian treatment of a cyclotomic field. 266 xii Preface This thesis originated from the idea of studying Zassenhaus's use of Hensel's lemma for a polynomial factorization algorithm. If this seems a far cry from the resulting study, consider that polynomial factorization over the p-adics is central to Hensel's version of algebraic number theory, in its role in the definition of prime divisors. Consider furthermore that Hensel's factorization algorithm demonstrated computability mainly in principle, whereas Zassenhaus was concerned with practical factorization. From this starting point, the history examined here presented itself. xiii Chapter 1 Introduction The subject of algebraic number theory, taught today with algebraic number fields Q(α) as the central objects, and with unique factorization recovered through the theory of ideals, has been built and rebuilt since the early nineteenth century in terms of different objects, and according to different methodologies. We review the nature of these alternative approaches here, and in the process will encounter the major questions that we will be concerned with in this thesis. In early work, such as P.G.L. Dirichlet's (1805 - 1859) studies on what we today call units in the rings of integers of number fields, 1 the notion was not that one was studying a collection called a \number field” but that one was simply studying the rational functions in a given algebraic number, i.e.
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    Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999.
  • Quadratic Forms Over Nonformally Real Fields with a Finite Number of Quaternion Algebras

    Quadratic Forms Over Nonformally Real Fields with a Finite Number of Quaternion Algebras

    Pacific Journal of Mathematics QUADRATIC FORMS OVER NONFORMALLY REAL FIELDS WITH A FINITE NUMBER OF QUATERNION ALGEBRAS CRAIG MCCORMACK CORDES Vol. 63, No. 2 April 1976 PACIFIC JOURNAL OF MATHEMATICS Vol. 63, No. 2, 1976 QUADRATIC FORMS OVER NONFORMALLY REAL FIELDS WITH A FINITE NUMBER OF QUATERNION ALGEBRAS CRAIG M. CORDES This paper is concerned with quadratic forms over a nonformally real field, F, of characteristic not two, which has only a finite number, m, of quaternion algebras. The number q— \F/F*\ is always assumed to be finite. Of central importance will be the radical, R, which can be defined by R = {aeF\G(l, -)a = F} where F=F-{0} and G(l, -a) is the set of nonzero elements represented by the form x2 — ay2 over F. The main result here is that a finite Witt ring (and hence the complete quadratic form structure) is deter- mined when m = 2 by the Witt group and the order of F/R. Other results include analyzing the quadratic form structure for fields which satisfy | F/R | <^ 8 and discovering an upper bound for m in terms of q and the largest index in F of all G(l, a). Finally, the concept of a quadratic form scheme is introduced. This idea comes from an attempt to abstract value sets of binary quadratic forms in order to eliminate the immediate necessity of a field in their study. Kaplansky [4] showed that if m = 2, the ^-invariant of a non- formally real field is four. Lam and Elman [3] demonstrated that the quaternion algebras form a subgroup of the Brauer group only when u = 1, 2, 4, or 8.