Advanced Algebra Along with a companion volume Basic Algebra
Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role
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Advisory Board Anthony W. Knapp, State University of New York at Stony Brook, Emeritus
Anthony W. Knapp
Advanced Algebra Along with a companion volume Basic Algebra
Birkh¨auser Boston • Basel • Berlin
Anthony W. Knapp 81 Upper Sheep Pasture Road East Setauket, NY 11733-1729 U.S.A. e-mail to: [email protected] http://www.math.sunysb.edu/˜ aknapp/books/a-alg.html
Cover design by Mary Burgess. Mathematics Subject Classicification (2000): 11-01, 13-01, 14-01, 16-01, 18G99, 55U99, 11R04, 11S15, 12F99, 14A05, 14H05, 12Y05, 14A10, 14Q99 Library of Congress Control Number: 2007936880 ISBN-13: 978-0-8176-4522-9
eISBN-13: 978-0-8176-4613-4
Basic Algebra Basic Algebra and Advanced Algebra (Set)
ISBN-13: 978-0-8176-3248-9 ISBN-13: 978-0-8176-4533-5
Printed on acid-free paper. c 2007 Anthony W. Knapp All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA) and the author, 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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(MP)
To Susan and To My Algebra Teachers: Ralph Fox, John Fraleigh, Robert Gunning, John Kemeny, Bertram Kostant, Robert Langlands, Goro Shimura, Hale Trotter, Richard Williamson
CONTENTS
Contents of Basic Algebra Preface List of Figures Dependence among Chapters Guide for the Reader Notation and Terminology
x xi xv xvi xvii xxi
I.
TRANSITION TO MODERN NUMBER THEORY 1. Historical Background 2. Quadratic Reciprocity 3. Equivalence and Reduction of Quadratic Forms 4. Composition of Forms, Class Group 5. Genera 6. Quadratic Number Fields and Their Units 7. Relationship of Quadratic Forms to Ideals 8. Primes in the Progressions 4n + 1 and 4n + 3 9. Dirichlet Series and Euler Products 10. Dirichlet’s Theorem on Primes in Arithmetic Progressions 11. Problems
II.
WEDDERBURN–ARTIN RING THEORY 1. Historical Motivation 2. Semisimple Rings and Wedderburn’s Theorem 3. Rings with Chain Condition and Artin’s Theorem 4. Wedderburn–Artin Radical 5. Wedderburn’s Main Theorem 6. Semisimplicity and Tensor Products 7. Skolem–Noether Theorem 8. Double Centralizer Theorem 9. Wedderburn’s Theorem about Finite Division Rings 10. Frobenius’s Theorem about Division Algebras over the Reals 11. Problems vii
1 1 8 12 24 31 35 38 50 56 61 67 76 77 81 87 89 94 104 111 114 117 118 120
viii
Contents
III. BRAUER GROUP 1. Definition and Examples, Relative Brauer Group 2. Factor Sets 3. Crossed Products 4. Hilbert’s Theorem 90 5. Digression on Co
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