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Algebra (Serge Lang).pdf
Cover
FOREWORD
Logical Prerequisites
CONTENTS
Part One THE BASIC OBJECTS OF ALGEBRA
CHAPTER1 Groups
1. MONOIDS
2. GROUPS
3. NORMAL SUBGROUPS
4. CYCLIC GROUPS
5. OPERATIONS OF A GROUP ON A SET
6. SYLOW SUBGROUPS
7. DIRECT SUMS AND FREE ABELIAN GROUPS
8. FINITELY GENERATED ABELIAN GROUPS
9. THE DUAL GROUP
10. INVERSE LIMIT AND COMPLETION
11. CATEGORIES AND FUNCTORS
12. FREE GROUPS
EXERCISES
CHAPTER2 Rings
1. RINGS AND HOMOMORPHISMS
2. COMMUTATIVE RINGS
3. POLYNOMIALS AND GROUP RINGS
4. LOCALIZATION
5. PRINCIPAL AND FACTORIAL RINGS
EXERC1SES
CHAPTER3 Modules
1. BASIC DEFINITIONS
2. THE GROUP OF HOMOMORPHISMS
3. DIRECT PRODUCTS AND SUMS OF MODULES
4. FREE MODULES
5. VECTOR SPACES
6. THE DUAL SPACE AND DUAL MODULE
7. MODULES OVER PRINCIPAL RINGS
8. EULER-POINCARE MAPS
9. THE SNAKE LEMMA
10. DIRECT AND INVERSE LIMITS
EXERCISES
CHAPTER4 Polynomials
1. BASIC PROPERTIES FOR POLYNOMIALS IN ONE VARIABLE
2. POLYNOMIALS OVER A FACTORIAL RING
3. CRITERIA FOR IRREDUCIBILITY
4. HILBERT'S THEOREM
5. PARTIAL FRACTIONS
6. SYMMETRIC POLYNOMIALS
7. MASON-STOTHERS THEOREM AND THE abc CONJECTURE
8. THE RESULTANT
9. POWER SERIES
EXERCISES
Part Two ALGEBRAIC EQUATIONS
CHAPTER5 Algebraic Extensions
1. FINITE AND ALGEBRAIC EXTENSIONS
2. ALGEBRAIC CLOSURE
3. SPLITTING FIELDS AND NORMAL EXTENSIONS
4. SEPARABLE EXTENSIONS
5. FINITE FIELDS
6. INSEPARABLE EXTENSIONS
EXERCISES
CHAPTER6 Galois Theory
1. GALOIS EXTENSIONS
2. EXAMPLES AND APPLICATIONS
3. ROOTS OF UNITY
4. LINEAR INDEPENDENCE OF CHARACTERS
5. THE NORM AND TRACE
6. CYCLIC EXTENSIONS
7. SOLVABLE AND RADICAL EXTENSIONS
8. ABELIAN KUMMER THEORY
9. THE EQUATION X^n-a=0
10. GALOIS COHOMOLOGY
11. NON-ABELIAN KUMMER EXTENSIONS
12. ALGEBRAIC INDEPENDENCE OF HOMOMORPHISMS
13. THE NORMAL BASIS THEOREM
14. INFINITE GALOIS EXTENSIONS
15. THE MODULAR CONNECTION
EXERCISES
CHAPTER7 Extensions of Rings
1. INTEGRAL RING EXTENSIONS
2. INTEGRAL GALOIS EXTENSIONS
3. EXTENSION OF HOMOMORPHISMS
EXERCISES
CHAPTER8 Transcendental Extensions
1. TRANSCENDENCE BASES
2. NOETHER NORMALIZATION THEOREM
3. LINEARLY DISJOINT EXTENSIONS
4. SEPARABLE AND REGULAR EXTENSIONS
5. DERIVATIONS
EXERCISES
CHAPTER9 Algebraic Spaces
1. HILBERT'S NULLSTELLENSATZ
2. ALGEBRAIC SETS,SPACES AND VARIETIES
3. PROJECTIONS AND ELIMINATION
4. RESULTANT SYSTEMS
5. SPEC OF A RING
EXERCISES
CHAPTER10 Noetherian Rings and Modules
1. BASIC CRITERIA
2. ASSOCIATED PRIMES
3. PRIMARY DECOMPOSITION
4. NAKAYAMA'S LEMMA
5. FILTERED AND GRADED MODULES
6. THE HILBERT POLYNOMIAL
7. INDECOMPOSABLE MODULES
EXERCISES
CHAPTER11 Real Fields
1. ORDERED FIELDS
2. REAL FIELDS
3. REAL ZEROS AND HOMOMORPHISMS
EXERCISES
CHAPTER12 Absolute Values
1. DEFINITIONS, DEPENDENCE,AND INDEPENDENCE
2. COMPLETIONS
3. FINITE EXTENSIONS
4. VALUATIONS
5. COMPLETIONS AND VALUATIONS
6. DISCRETE VALUATIONS
7. ZEROS OF POLYNOMIALS IN COMPLETE FIELDS
EXERCISES
Part Three LINEAR ALGEBRA and REPRESENTATIONS
CHAPTER13 Matrices and Linear Maps
1. MATRICES
2. THE RANK OF A MATRIX
3. MATRICES AND LINEAR MAPS
4. DETERMINANTS
5. DUALITY
6. MATRICES AND BILINEAR FORMS
7. SESQUILINEAR DUALITY
8. THE SIMPLICITY OF SL2(F)/+-1
9. THE GROUPS Ln(F),n>=3.
EXERCISES
CHAPTER14 Representation of One Endomorphism
1. REPRESENTATIONS
2. DECOMPOSITION OVER ONE ENDOMORPHISM
3. THE CHARACTERISTIC POLYNOMIAL
EXERCISES
CHAPTER15 Structure of Bilinear Forms
1. PRELIMINARIES,ORTHOGONAL SUMS
2. QUADRATIC MAPS
3. SYMMETRIC FORMS, ORTHOGONAL BASES
4. SYMMETRICFORMS OVER ORDERED FIELDS
5. HERMITIAN FORMS
6. THE SPECTRAL THEOREM(HERMITIAN CASE)
7. THE SPECTRAL THEOREM(SYMMETRIC CASE)
8. ALTERNATING FORMS
9. THE PFAFFIAN
10. WITT'S THEOREM
EXERCISES
CHAPTER16 The Tensor Product
1. TENSOR PRODUCT
2. BASIC PROPERTIES
3. FLAT MODULES
4. EXTENSION OF THE BASE
5. SOME FUNCTORIAL ISOMORPHISMS
6. TENSOR PRODUCT OF ALGEBRAS
7. THE TENSOR ALGEBRA OF A MODULE
8. SYMMETRIC PRODUCTS
EXERCISES
CHAPTER17 Semisimplicity
1. MATRICES AND LINEAR MAPSOVERNON-COMMUTATIVE RINGS
2. CONDITIONS DEFINING SEMISIMPLICITY
3. THE DENSITY THEOREM
4. SEMISIMPLE RINGS
5. SIMPLE RINGS
6. THE JACOBSON RADICAL,BASE CHANGE,AND TENSOR PRODUCTS
7. BALANCED MODULES
EXERCISES
CHAPTER18 Representations of Finite Groups
1. REPRESENTATIONS AND SEMISIMPLICITY
2. CHARACTERS
3. 1-DIMENSIONAL REPRESENTATIONS
4. THE SPACE OF CLASS FUNCTIONS
5. ORTHOGONALITY RELATIONS
6. INDUCED CHARACTERS
7. INDUCED REPRESENTATIONS
8. POSITIVE DECOMPOSITION OF THE REGULAR CHARACTER
9. SUPERSOLVABLE GROUPS
11. FIELD OF DEFINITION OF A REPRESENTATION
12. EXAMPLE:GL2 OVER A FINITE FIELD
EXERCISES
CHAPTER19 The Alternating Product
1. DEFINITION AND BASIC PROPERTIES
2. FITTING IDEALS
3. UNIVERSAL DERIVATIONS AND THE DE RHAM COMPLEX
4. THE CLIFFORD ALGEBRA
EXERCISES
Part Four HOMOLOGICAL ALGEBRA
CHAPTER20 General Homology Theory
1. COMPLEXES
2. HOMOLOGY SEQUENCE
3. EULER CHARACTERISTIC AND THE GROTHENDIECK GROUP
4. INJECTIVE MODULES
5. HOMOTOPIES OF MORPHISMS OF COMPLEXES
6. DERIVED FUNCTORS
7. DELTA-FUNCTORS
8. BIFUNCTORS
9. SPECTRAL SEQUENCES
EXERCISES
CHAPTER21 Finite Free Resolutions
1. SPECIAL COMPLEXES
2. FINITE FREE RESOLUTIONS
3. UNIMODULAR POLYNOMIAL VECTORS
4. THE KOSZUL COMPLEX
EXERCISES
APPENDIX1 The Transcendence of e and π
APPENDIX2 Some Set Theory
1. DENUMERABLE SETS
2. ZORN'S LEMMA
3. CARDINAL NUMBERS
4. WELL-ORDERING
EXERCISES
Bibliography
INDEX
Graduate Texts in Mathematics
Back Cover
Graduate Texts in Mathematics 211
S. Axler
Editorial Board
F.W. Gehring K.A. Ribet
Springer
New York
Berlin
Heidelberg
Barcelona
Hong Kong
London
Milan
Paris
Singapore
Tokyo
Serge Lang
Department of Mathematics
Yale University
New Haven, CT 96520
USA
Editorial Board
S. Axler
Mathematics Department
San Francisco State
University
San Francisco, CA 94132
USA
F. W. Gehring
Mathematics Department
East Hall
University of Michigan
Ann Arbor, MI 48109
USA
K.A. Ribet
Mathematics Department
University of California
at Berkeley
Berkeley, CA 94720-3840
USA
Mathematics Subject Classification (2000): 13-01, 15-01, 16-01, 20-01
Library of Congress Cataloging-in-Publication Data
Algebra I Serge Lang.-Rev. 3rd ed.
p.
em. - (Graduate texts in mathematics; 211)
Includes bibliographical references and index.
ISBN 0-387-95385-X (alk. paper)
1. Algebra.
II. Series.
I. Title.
QA 154.3.L3
512--dc21
2002
Printed on acid-free paper.
2001054916
This title was previously published by Addison-Wesley, Reading, MA 1993.
<9 2002 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written pennission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York,
NY 10010, 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 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.
Production managed by Terry Kornak; manufactunng supervised by Erica Bresler.
Revisions typeset by Asco Typesetters, North Point, Hong Kong.
Printed and bound by Edwards Brothers, Inc., Ann Arbor, MI.
Printed in the United States of America.
9 8 7 6 5 432 1
ISBN 0-387 -95385-X
SPIN 10855619
Springer-Verlag New York Berlin Heidelberg
A member of BertelsmannSp ringer Science+Business Media GmbH
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
in all of mathematics:
As I see it, the graduate course in algebra must primarily prepare students
to handle the algebra which they will meet
topology,
partial differential equations, differential geometry, algebraic geometry, analysis,
and representation theory, not to speak of algebra itself and algebraic number
theory with all its ramifications. Hence I have inserted throughout references to
papers and books which have appeared during 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,
especially with examples from other branches of mathematics, of necessity some
terms may not be defined, or may be defined only later in the book. I have tried
to help the reader not 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 number of exercises. On the whole, I have tried to make
I
the exercises complement the examples, and to give them aesthetic appeal.
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, a book must be projected in a totally ordered way on the 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 several parts. The first part introduces the 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.
There is some sporadic feedback between them, but their unification takes place
at the next level of mathematics, which is suggested, for instance,
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 of the algebraic closure in groups of matrices (the
linear approach), and giving an explicit determination of the irrationalities gen-
erating algebraic extensions (the equations approach). At the moment, repre-
sentations in GL 2 are at the center of attention from various quarters, and readers
I have
will see GL 2 appear several times throughout the book. For instance,
found it appropriate to add a section describing all
irreducible characters of
GL 2(F) when F is a finite field. Ultimately, GL 2 will appear as the simplest but
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.
in
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.
Algebraic geometry having progressed in many ways, it is now sometimes return-
ing to older and harder problems, such as searching for the effective construction
of polynomials vanishing on certain algebraic sets, and the older elimination
procedures of last century serve as an introduction to those problems.
Except for 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, some topics have been reordered. I was informed by readers and review-
ers of the tension existing between having a textbook usable for relatively inex-
perienced students, and a reference book where results could easily be found in
a systematic arrangement. I have tried to reduce this tension by moving all the
homological algebra to a fourth part, and by integrating the commutative algebra
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 as to allow maximal flexibility in this respect, and
I have frequently committed the crime of lese- Bourbaki by repeating short argu-
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
basic course, there exist several options for leading the course in various direc-
tions. It is impossible to treat all of them with the same degree of thoroughness.
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 agreement on these matters, not even between the
author and himself. Thus the concrete decisions as to what to include and what
not to incl ude are finally taken on grounds of general coherence and aesthetic
balance. Anyone teaching the course will want,to impress their own personality
on the material, and may push certain 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
algebra required a substantial increase in size from the first to the second edition,
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, an vector spaces. Since the Galois theory
gives very quickly an impression of dpth, this is very satisfactory in many
respects.
It is appropriate here to recall my original 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 aspects of 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, the amount of algebra which one should ideally absorb during
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 lecturer during a one-year course. 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 easier to use as
I have tried to expand the general mathematical
a text. By these additions,
perspective of the reader, insofar as algebra relates to other parts of mathematics.
Acknowledgements
I am indebted to many people who have contributed comments and criticisms
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. Finally I thank the many
people who have made suggestions and corrections, especially George Bergman,
Chee-Whye Chin, Ki-Bong Nam, David Wasserman, and Randy Scott, who
provided me with a list of errata. I also thank Thomas Shiple and Paul Vojta
for their lists of errata to the third edition. These have been corrected in the
subsequent printings.
Serge Lang
New Haven
For the 2002 and beyond Springer printings
From now on, Algebra appears with Springer-Verlag, like the rest of my
books. With this change, I considered the possibility of a new edition, but de-
cided against it. I view the book as very stable. The only addition which I
would make, if starting from scratch, would be some of the algebraic properties
of SLn and GLn (over R or C), beyond the proof of simplicity in Chapter XIII.
As things stood, I just inserted some exercises concerning some aspects which
everybody should know. Readers can see these worked out in Jorgenson/Lang,
Spherical Inversion on SLn(R), Springer Verlag 2001, as well as other basic
algebraic properties on which analysis is superimposed so that algebra in this
context appears as a supporting tool.
I thank specifically Tom von Foerster, Ina Lindeman and Mark Spencer for
their editorial support at Springer, as well as Terry Kornak and Brian Howe
whc have taken care of production.
Serge Lang
New Haven 2002
Logical Prerequisites
We assume that the reader is familiar with sets, and with the symbols n, U,
::>, C, E. If A, B are sets, we use the symbol A C B to mean that A is contained
in B but may be equal to B. Similarly for A ::> B.
If f: A
B is a mapping of one set into another, we write
x
f(x)
Let f: A
to denote the effect of f on an element x of A. We distinguish between the
arrows and. We denote by f(A) the set of all elementsf(x), with x E A.
B be a mapping (also called a map). We say that f is injective
if x =F y implies f(x) =F f(y). We say f is surjective if given b E B there exists
a E A such that f(a) = b. We say that f is bijective if it is both surjective and
injective.
.
A subset A of a set B is said to be proper if A =F B.
Let f: A
B be a map, and A' a subset of A. The restriction of f to A'
is
a map of A' into B denoted by f I A'.
If f: A
Band g : B -+ C are maps, then we have a composite map g 0 f
such that (g 0 f)(x) = g(f(x)) for all x E A.
Letf: A
B be a map, and B' a subset of B. Byf - l(B') we mean the subset
of A consisting of all x E A such that f(x) E B'. We call it the inverse image of
B'. We call f(A) the image of f.
A diagram
A
f) B
\)
C
is said to be commutative if 9 0 f = h. Similarly, a diagram
A
]
C
f) B
]g
) D
'"
ix
X
LOGICAL PREREQUISITES
is said to be commutative if g 0 f = t/J 0 qJ. We deal sometimes with more
complicated diagrams, consisting of arrows between various objects. Such
diagrams are called commutative if, whenever it is possible to go from one
object to another by means of two sequences of arrows, say
and
then
A I
II
)
A
2
12
) ...
In-I
) An
Al
91
B)
2
92
) ...
9m-1
) Bm = An'
In-I 0 ·
·
· 0 II = 9m-1 0 ·
·
· 0 91'
in other words, the composite maps are equal. Most of our diagrams are
composed of triangles or squares as above, and to verify that a diagram con-
sisting of triangles or squares is commutative, it suffices to verify that each
triangle and square in it is commutative.
We assume that the reader is acquainted with the integers and rational
numbers, denoted respectively by Z and Q. For many of our examples, we also
assume that the reader knows the real and complex numbers, denoted. by R
and C.
Let A and I be two sets. By a family of elements of A, indexed by I, one
means a map f: I -. A. Thus for each i E I we are given an element f(i) E A.
Although a family does not differ from a map, we think of it as determining a
collection of objects from A, and write it often as
or
{f(i)}iel
{aJ i e I'
writing ai instead of f(i). We call I the indexing set.
We assume that the reader knows what an equivalence relation is. Let A
be a set with an equivalence relation, let E be an equivalence class of elements
of A. We sometimes try to define a map of the equivalence classes into some
set B. To define such a map f on the class E, we sometimes first give its value
on an element x E E (called a representative of E), and then show that it is
independent of the choice of representative x E E.
In that case we say that f
is well defined.
We have products of sets, say finite products A x B, or At x ... x An' and
prod ucts of families of sets.
We shall use Zorn's lemma, which we describe in Appendix 2.
We let #(S) denote the number of elements of a set S, also called the
cardinality of S. The notation is usually employed when S is finite. We also
write #(S) = card(S).
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