Number Theory An Introduction to Mathematics
"Number Theory" is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included. The book i
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W.A. Coppel
Number Theory An Introduction to Mathematics Second Edition
W.A. Coppel 3 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.
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For Jonathan, Nicholas, Philip and Stephen
Contents
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Part A I
The Expanding Universe of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 Sets, Relations and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Integers and Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Quaternions and Octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Vector Spaces and Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . 10 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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