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Undergraduate Texts m Mathematics Editors S. Axler F. W. Gehring K. A. Ribet Springer Science+Business Media, LLC Springer Books on Elementary Mathematics by Serge Lang MATH! Encounters with High School Students 1985, ISBN 96129-1 The Beauty of Doing Mathematics 1985, ISBN 96149-6 Geometry. A High School Course (with G. Murrow), Second Edition 1989, ISBN 96654-4 Basic Mathematics 1988, ISBN 96787-7 A First Course in Calculus 1986, ISBN 96201-8 Calculus of Several Variables 1987, ISBN 96405-3 Introduction to Linear Algebra 1986, ISBN 96205-0 Linear Algebra 1987, ISBN 96412-6 Undergraduate Algebra, Second Edition 1990, ISBN 97279-X Undergraduate Analysis 1983, ISBN 90800-5 Complex Analysis 1985, ISBN 96085-6 Serge Lang Und ergra duat e Algebra Second Edition 'Spring er Serge Lang Department of Mathematics Yale University New Haven, CT 06520 USA Editorial Board S. Axler F.W. Gehring Department of Mathematics San Francisco State University San Francisco, CA 94132 USA Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA K.A. Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA AMS Subject Classification (1980): 13-01, 15-01 Library of Congress Cataloging-in-Publication Data Lang, Serge, 1927. Undergraduate algebra / Serge Lang. _2nd ed. p. cm.-(Undergraduate texts in mathematics) Includes index. ISBN 978-1-4757-6900-5 ISBN 978-1-4757-6898-5 (eBook) DOI 10.1007/978-1-4757-6898-5 \. Algebra. I. Title. II. Undergraduate texts in mathematics (Springer-Verlag). QAI52.2.L36 1990 512.9-dc20 90-9648 Printed on acid-free paper. © 1987, 1990 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1990 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. 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not epecially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may be accordingly used freely by anyone. Typeset by Techset Composition Limited, Salisbury, Wilts. England. 9 8 7 6 5 4 (Corrected fourth printing, 2001) ISBN 978-1-4757-6900-5 SPIN 10838227 Foreword This book, together with Linear Algebra, constitutes a curriculum for an algebra program addressed to undergraduates. The separation of the linear algebra from the other basic algebraic structures fits all existing tendencies affecting undergraduate teaching, and I agree with these tendencies. I have made the present book self contained logically, but it is probably better if students take the linear algebra course before being introduced to the more abstract notions of groups, rings, and fields, and the systematic development of their basic abstract properties. There is of course a little overlap with the book Linear Algebra, since I wanted to make the present book self contained. I define vector spaces, matrices, and linear maps and prove their basic properties. The present book could be used for a one-term course, or a year's course, possibly combining it with Linear Algebra. I think it is important to do the field theory and the Galois theory, more important, say, than to do much more group theory than we have done here. There is a chapter on finite fields, which exhibit both features from general field theory, and