Graduate Texts in Mathematics Editorial Board S. Axler F.W. Gehring K.A. Ribet BOOKS OF RELATED INTEREST BY SERGE LANG Math Talks for Undergraduates 1999, ISBN 0-387-98749-5 Linear Algebra, Third Edition 1987, ISBN 0-387-96412-6 Undergraduate Algebra, Second Edition 1990, ISBN 0-387-97279-X Undergraduate Analysis, Second Edition 1997, ISBN 0-387-94841-4 Complex Analysis, Third Edition 1993, ISBN 0-387-97886 Real and Functional Analysis, Third Edition 1993, ISBN 0-387-94001-4 Algebraic Number Theory, Second Edition 1994, ISBN 0-387-94225-4 OTHER BOOKS BY LANG PUBLISHED BY SPRINGER-VERLAG Introduction to Arakelov Theory • Riemann-Roch Algebra (with William Fulton) • Complex Multiplication • Introduction to Modular Forms • Modular Units (with Daniel Kubert) • Fundamentals of Diophantine Geometry • Elliptic Functions • Number Theory III • Survey of Diophantine Geometry • Fundamentals of Differential Geometry • Cyclotomic Fields I and II • SL(R) • Abelian Varieties • 2 Introduction to Algebraic and Abelian Functions • Introduction to Diophantine Approximations • Elliptic Curves: Diophantine Analysis • Introduction to Linear Algebra • Calculus of Several Variables • First Course in Calculus • Basic Mathematics • Geometry: A High School Course (with Gene Murrow) • Math! Encounters with High School Students • The Beauty of Doing Mathematics • THE FILE • CHALLENGES Serge Lang Algebra Revised Third Edition Springer Serge Lang Department of Mathematics Yale University New Haven, CT 96520 USA Editorial Board S. Axler Mathematics Department F.W. Gehring K.A. Ribet San Francisco State Mathematics Department Mathematics Department University East Hall University of California, San Francisco, CA 94132 University of Michigan Berkeley USA Ann Arbor, MI 48109 Berkeley, CA 94720-3840 [email protected] USA USA [email protected]. [email protected] umich.edu Mathematics Subject Classification (2000): 13-01, 15-01, 16-01, 20-01 Library of Congress Cataloging-in-Publication Data Algebra /Serge Lang.—Rev. 3rd ed. p. cm.—(Graduate texts in mathematics; 211) Includes bibliographical references and index. ISBN 978-1-4612-6551-1 ISBN 978-1-4613-0041-0 (eBook) DOI 10.1007/978-1-4613-0041-0 1. Algebra. I. Title. II. Series. QA154.3.L3 2002 512—dc21 2001054916 ISBN 978-1-4612-6551-1 Printed on acid-free paper. This title was previously published by Addison-Wesley, Reading, MA 1993. © 2002 Springer Science+Business Media New York Originally published by Springer Science+Business Media LLC in 2002 Softcover reprint of the hardcover 3rd edition 2002 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 especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. 9 8 (corrected printing 2005) springer.com FOREWORD The present book is meant as a basic text for a one-year course in algebra, at the graduate level. A perspective on algebra As I see it, the graduate course in algebra must primarily prepare students to handle the algebra which they will meet in all of mathematics: topology, partialdifferentialequations,differentialgeometry,algebraicgeometry,analysis, and representation theory, not to speak of algebra itself and algebraic number theory with all its ramifications. Hence Ihave inserted throughout references to papersand bookswhich have appearedduring the last decades,to indicate some of the directions in which the algebraic foundations provided by this book are used; I have accompanied these references with some motivating comments, to explain how the topics of the present book fit into the mathematics that is to come subsequently in various fields; and I have also mentioned some unsolved problems of mathematics in algebra and number theory. The abc conjecture is perhaps the most spectacular of these. Often when such comments and examples occur out of the logical order, especiallywithexamplesfrom otherbranchesofmathematics,ofnecessitysome terms may not be defined, or may be defined only laterin the book.I have tried to help the readernot only by making cross-references within the book, but also by referring to other books or papers which I mention explicitly. I have also added a numberof exercises. On the whole, I have tried to make the exercises complement the examples, and to give them aesthetic appeal. I have tried to use the exercises also to drive readers toward variations and appli cations of the main text, as well as toward working out special cases, and as openings toward applications beyond this book. Organization Unfortunately, abook must beprojectedinatotallyorderedway onthe page axis, but that's not the way mathematics "is", so readers have to make choices how to reset certain topics in parallel for themselves, rather than in succession. v vi FOREWORD I have inserted cross-references to help them do this, but different people will make different choices at different times depending on different circumstances. The book splits naturally into severalparts. The firstpartintroducesthe basic notions of algebra. After these basic notions, the book splits in two major directions: the direction of algebraic equations including the Galois theory in Part II; and the direction of linear and multilinear algebra in Parts III and IV. Thereis some sporadic feedback between them,but their unification takes place at the next level of mathematics, which is suggested, for instance, in §15 of Chapter VI. Indeed, the study of algebraic extensions of the rationals can be carried out from two points of view which are complementary and interrelated: representing the Galois group ofthe algebraic closure ingroups of matrices (the linear approach), and giving an explicitdetermination of the irrationalities gen erating algebraic extensions (the equations approach). At the moment, repre sentationsinGL areatthecenterofattentionfrom various quarters, and readers 2 will see GL appear several times throughout the book. For instance, I have 2 found it appropriate to add a section describing all irreducible characters of GL when F is a finite field. Ultimately, GL will appearas the simplest but 2(F) 2 typical case of groups of Lie types, occurring both in a differential context and over finite fields or more general arithmetic rings for arithmetic applications. After almost a decade since the second edition, I find that the basic topics of algebra have become stable, with one exception. I have added two sections on elimination theory, complementing the existing section on the resultant. Algebraicgeometryhaving progressedinmanyways, itisnowsometimesreturn ingtoolderandharder problems, such assearchingfortheeffectiveconstruction of polynomials vanishing on certain algebraic sets, and the older elimination procedures of last century serve as an introduction to those problems. Exceptfor this addition, the main topics of the book are unchanged from the second edition, but I have tried to improve the book in several ways. First,sometopicshavebeenreordered.Iwasinformedbyreaders andreview ers ofthe tension existing between having atextbook usable for relatively inex perienced students, and areference book where results could easily be found in a systematic arrangement. I have tried to reduce this tension by moving all the homologicalalgebratoafourth part, andbyintegratingthecommutativealgebra with the chapter on algebraic sets and elimination theory, thus giving an intro duction to different points of view leading toward algebraic geometry. The book as a text and a reference In teaching the course, one might wish to push into the study of algebraic equations through Part II, or one may choose to go first into the linear algebra of Parts III and IV. One semester could be devoted to each, for instance. The chapters have been so written astoallow maximal flexibility inthis respect, and Ihave frequently committedthecrime oflese-Bourbaki byrepeating shortargu ments or definitions to make certain sections or chapters logically independent of each other. FOREWORD vii Granting the material which under no circumstances can be omitted from a basiccourse, there exist severaloptions for leading the course in various direc tions. It is impossible totreat allof them withthe samedegree ofthoroughness. The precise point at which one is willing to stop in any given direction will depend on time, place, and mood. However, any book with the aims of the present one must include a choice of topics, pushing ahead-in deeper waters, while stopping short of full involvement. There can be no universal agreementon these matters, not even between the author and himself. Thus the concrete decisions as to what to include and what not to include are finally taken on grounds of general coherence and aesthetic balance. Anyone teaching the course will want to impresstheirown personality on the material, and may pushcertain topics with more vigor than I have, at the expense of others. Nothing in the present book is meant to inhibit this. Unfortunately, the goal to present a fairly comprehensive perspective on algebrarequiredasubstantial increaseinsizefrom thefirsttothe secondedition, and a moderate increase in this third edition. These increases require some decisions as to what to omit in a given course. Many shortcuts can be taken in the presentation of the topics, which admits many variations. For instance, one can proceed into field theory and Galois theory immediately after giving the basic definitions for groups, rings, fields,polynomials in one variable,and vector spaces. Since the Galois theory gives very quickly an impression of depth, this is very satisfactory in many respects. It isappropriate here to recall myoriginal indebtedness to Artin,who first taught me algebra. The treatment of the basics of Galois theory is much influenced by the presentation in his own monograph. Audience and background As I already stated in the forewords of previous editions, the present book is meant for the graduate level, and I expect most of those coming to it to have had suitable exposure to some algebra in an undergraduate course, or to have appropriate mathematical maturity. I expect students taking a graduate course to have had some exposure to vector spaces, linear maps, matrices, and they will no doubt have seen polynomials at the very least in calculus courses. My books Undergraduate Algebra and Linear Algebra provide more than enough background for a graduate course. Such elementary texts bring out in parallel the two basic aspectsof algebra, and are organized differently from the present book, where both aspects are deepened. Of course, some aspects of the linear algebra in Part III of the present book are more "elementary" than some aspects of Part II, which deals with Galois theory and the theory of polynomial equations in several variables. Because Part II has gone deeper into the study of algebraic equations, of necessity the parallel linear algebra occurs only later in the total ordering of the book. Readers should view both parts as running simultaneously. viii FOREWORD Unfortunately, theamount ofalgebrawhich oneshould ideallyabsorbduring this first year in order to have a proper background (irrespective of the subject in which one eventually specializes) exceeds the amount which can be covered physically by a lecturerduring aone-yearcourse. Hence more material must be included than can actually be handled in class. I find it essential to bring this material to the attention of graduate students. I hope that the various additions and changes make the book easierto use as a text. By these additions, I have tried to expand the general mathematical perspectiveofthereader, insofarasalgebrarelatestootherpartsofmathematics. Acknowledgements Iam indebtedtomanypeople whohavecontributedcommentsandcriticisms for the previous editions, but especially to Daniel Bump, Steven Krantz, and Diane Meuser, who provided extensive comments as editorial reviewers for Addison-Wesley. I found their comments very stimulating and valuable in pre paring this third edition. I am much indebted to Barbara Holland for obtaining these reviews when she was editor. I am also indebted to Karl Matsumoto who supervised production under very trying circumstances. I thank the many peo ple who have made suggestions and corrections, especially George Bergman and students in his class, Chee-Whye Chin, Ki-Bong Nam, David Wasserman, Randy Scott, Thomas Shiple, Paul Vojta,Bjorn Poonen andhisclass,in partic ular Michael Manapat. For the 2002 and beyondSpringer printings From now on, Algebra appears with Springer-Verlag, like the rest of my books. With this change, I considered the possibility ofa new edition, but de cided against it. I view the book as very stable. The only addition which I would make,ifstartingfrom scratch,would besome ofthe algebraicproperties ofSLnand GLn(over R or C), beyond the proofofsimplicity in ChapterXIII. As things stood, I just inserted some exercises concerning some aspects which everybody should know. The material actually isnow inserted in a new edition ofUndergraduate Algebra,where it properly belongs. The algebra appears as a supporting tool for doing analysis on Lie groups, cf. for instance Jorgenson/ Lang SphericalInversion on SLn(R), SpringerVerlag 2001. I thank specifically Tom von Foerster, Ina Lindemann and Mark Spencer for their editorial support at Springer, as well as Terry Kornak and Brian Howe who have taken care ofproduction. Serge Lang New Haven 2004 Logical Prerequisites We assume that the reader isfamiliar with sets, and with the symbols n, U, ~, C, E. IfA,B are sets, we use the symbol A C B to mean that A iscontained inB but may be equal to B. Similarly for A ~ B. Iff :A -> B isa mapping ofone set into another, wewrite X1---+f(x) to denote the effect off on an element x of A. We distinguish between the arrows -> and 1---+. We denote byf(A) the set ofall elementsf(x), with x E A. Let f :A -> B be a mapping (also called a map). We say that f is injective ifx # y implies f(x) # f(y). We say f is surjective ifgiven be B there exists aE A such that f(a) = b. We say that f is bijective ifit is both surjective and injective. Asubset A ofa set B issaid to be proper ifA # B. Let f :A -> B be a map, and A' a subset of A. The restriction off to A' is a map of A' into Bdenoted byfIA'. Iff :A -> Band 9:B -> C are maps, then we have a composite map 9 0 f such that (g 0 f)(x) = g(f(x» for all x E A. Letf:A -> B bea map,and B'a subsetofB. Byf- 1(B')wemean the subset ofA consisting ofall x E A such that f(x) E B'. We call it the inverseimageof B'. Wecall f(A) the imageoff. Adiagram C issaid to becommutative ifg of = h. Similarly, a diagram A~B •j j. C---->D '" ix