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Part 1 book “Number theory - An introduction to mathematics” has contents: The expanding universe of numbers, divisibility, more on divisibility, continued fractions and their uses, hadamard’s determinant problem, hensel’s p-adic numbers.
Universitext For other titles in this series, go to www.springer.com/series/223 W.A Coppel Number Theory An Introduction to Mathematics Second Edition W.A Coppel Jansz Crescent 2603 Griffith Australia Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczy´nski, Case Western Reserve University ISBN 978-0-387-89485-0 e-ISBN 978-0-387-89486-7 DOI 10.1007/978-0-387-89486-7 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009931687 Mathematics Subject Classification (2000): 11-xx, 05B20, 33E05 © c Springer Science+ Business Media, LLC 2009 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+ Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) For Jonathan, Nicholas, Philip and Stephen Contents Preface to the Second Edition xi Part A I The Expanding Universe of Numbers Sets, Relations and Mappings Natural Numbers Integers and Rational Numbers Real Numbers Metric Spaces Complex Numbers Quaternions and Octonions Groups Rings and Fields Vector Spaces and Associative Algebras 10 Inner Product Spaces 11 Further Remarks 12 Selected References Additional References 1 10 17 27 39 48 55 60 64 71 75 79 82 II Divisibility Greatest Common Divisors The B´ezout Identity Polynomials Euclidean Domains Congruences Sums of Squares Further Remarks Selected References Additional References 83 83 90 96 104 106 119 123 126 127 viii Contents III More on Divisibility The Law of Quadratic Reciprocity Quadratic Fields Multiplicative Functions Linear Diophantine Equations Further Remarks Selected References Additional References 129 129 140 152 161 174 176 178 IV Continued Fractions and Their Uses The Continued Fraction Algorithm Diophantine Approximation Periodic Continued Fractions Quadratic Diophantine Equations The Modular Group Non-Euclidean Geometry Complements Further Remarks Selected References Additional References 179 179 185 191 195 201 208 211 217 220 222 V Hadamard’s Determinant Problem What is a Determinant? Hadamard Matrices The Art of Weighing Some Matrix Theory Application to Hadamard’s Determinant Problem Designs Groups and Codes Further Remarks Selected References 223 223 229 233 237 243 247 251 256 258 VI Hensel’s p-adic Numbers Valued Fields Equivalence Completions Non-Archimedean Valued Fields Hensel’s Lemma Locally Compact Valued Fields Further Remarks Selected References 261 261 265 268 273 277 284 290 290 Contents ix Part B VII The Arithmetic of Quadratic Forms Quadratic Spaces The Hilbert Symbol The Hasse–Minkowski Theorem Supplements Further Remarks Selected References 291 291 303 312 322 324 325 VIII The Geometry of Numbers Minkowski’s Lattice Point Theorem Lattices Proof of the Lattice Point Theorem; Other Results Voronoi Cells Densest Packings Mahler’s Compactness Theorem Further Remarks Selected References Additional References 327 327 330 334 342 347 352 357 360 362 IX The Number of Prime Numbers Finding the Problem Chebyshev’s Functions Proof of the Prime Number Theorem The Riemann Hypothesis Generalizations and Analogues Alternative Formulations Some Further Problems Further Remarks Selected References Additional References 363 363 367 370 377 384 389 392 394 395 398 X A Character Study Primes in Arithmetic Progressions Characters of Finite Abelian Groups Proof of the Prime Number Theorem for Arithmetic Progressions Representations of Arbitrary Finite Groups Characters of Arbitrary Finite Groups Induced Representations and Examples Applications Generalizations Further Remarks 10 Selected References 399 399 400 403 410 414 419 425 432 443 444 ... University ISBN 97 8-0 -3 8 7-8 948 5-0 e-ISBN 97 8-0 -3 8 7-8 948 6-7 DOI 10 .10 07/97 8-0 -3 8 7-8 948 6-7 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 20099 316 87 Mathematics Subject... 17 9 17 9 18 5 19 1 19 5 2 01 208 211 217 220 222 V Hadamard’s Determinant Problem What is a Determinant? ... Coppel, Number Theory: An Introduction to Mathematics, Universitext, DOI: 10 .10 07/97 8-0 -3 8 7-8 948 6-7 _1, © Springer Science + Business Media, LLC 2009 I The Expanding Universe of Numbers We will not