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ALGEBRA Michael Artin Massachusetts Institute of Technology 1) UP | COLLEGE OF SCIENCE DILIMAN | CENTRAL LIBRARY E UDSCB0035140 PRENTICE HALL Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Artin, Michael. Algebra / Michael Artin. p em. Includes bibliographical references and index. ISBN 0-13-004763-5 1. Algebra. E Title. QAI54.2.A77 1991 512.9-—de20 Figure 4.16 from Zeitschrifi fir Kristallographie Editorial/Produetion Supervision and Interior Design: Ruth Cottrel] ELA Prepress Buyer: Paula Massenaro Manufacturing Buyer: Lori Bulwin APAE & 1991 by Prentice-Hall, Inc. = A Simon & Schuster Company Upper Saddle River, New Jersey 07458 AL rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Printed in the United States of America 0 98 SE ISBN 0-33-DD4y?b3-5 9 d78 Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc. Toronto Prentice-Hall Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro 91-2107 20000> 301047632 || | Contents Chapter 1 Chapter 2 Preface xiti A Note for the Teacher xv Matrix Operations 1 1. The Basic Operations 1 2. Row Reduction 9 3. Determinants 18 4. Permutation Matrices 24 5. Cramer's Rule 28 EXERCISES 31 Groups 38 1. The Definition of a Group 38 2. Subgroups 44 3. Isomorphisms 48 4. Homomorphisms 51 5. Equivalence Relations and Partitions 53 6. Cosets 57 7. Restriction of a Homomorphism to a Subgroup 59 8. Products of Groups 61 9. Modular Arithmetic 64 16. Quotient Groups 66 EXERCISES 69 vii Chapter 3 Chapter 4 Chapter 5 Chapter 6 Vector Spaces 1. Real Vector Spaces 78 Abstract Fields 82 Bases and Dimension 87 Computation with Bases 94 Infinite-Dimensional Spaces 100 Direct Sums 102 EXERCISES 104 anagmpn Linear Transformations The Dimension Formula 109 The Matrix of a Linear Transformation 111 Linear Operators and Eigenvectors 115 The Characteristic Polynomial 120 Orthogonal Matrices and Rotations 123 Diagonalization 130 Systems of Differential Equations 133 The Matrix Exponential 138 EXERCISES 145 Mana nm Symmetry 1. Symmetry of Plane Figures 155 The Group oí Motions of the Plane 157 Finite Groups of Motions 162 Discrete Groups of Motions 166 Abstract Symmetry: Group Operations 175 The Operation on Cosets 178 The Counting Formula 180 Permutation Representations 182 Finite Subgroups of the Rotation Group 184 EXERCISES 188 cpa ana tn More Group Theory 1. The Operations of a Group on Itself 197 2. The Class Equation of the Icosahedral Group 200 3. Operations on Subsets 203 Contents 78 109 155 197 7. 8. 9. 10. Contents Characters 316 Permutation Representations and the Regular Representation 321 The Representations of the Icosahedral Group 323 One-Dimensional Representations 325 Schur's Lemma, and Proof of the Orthogonality Relations 325 Representations of the Group SU> 330 EXERCISES 335 Chapter 10 Rings Chapter 11 1. paga Definition of a Ring 345 Formal Construction of Integers and Polynomials 347 Homomorphisms and Ideals 353 Quotient Rings and Relations in a Ring 359 Adjunction of Elements 364 Integral Domains and Fraction Fields 368 Maximal Ideals 370 Algebraic Geometry 373 EXERCISES 379 Factorization 1. Factorization of Integers and Polynomials 389 2. Unique Factorization Domains, Principal Ideal Domains, and Euclidean Domains 392 3. Gauss's Lemma 398 4. Explicit Factorization of Polynomials 402 5. Primes in the Ring of Gauss Integers 406 6. Algebraic Integers 409 7. Factorization in Imaginary Quadratic Fields 414 8. Ideal Factorization 419 9. The Relation Between Prime Ideals of R and Prime Integers 424 10. Ideal Classes in Imaginary Quadratic Fields 425 11. Reai Quadratic Fields 433 345 389 Contents Chapter 12 Chapter 13 Chapter 14 12. Some Diophantine Equations 437 EXERCISES 440 Modules sas ps The Definition of a Module 450 Matrices, Free Modules, and Bases 452 The Principle of Permanence of Identities 456 Diagonalization of Integer Matrices 457 Generators and Relations for Modules 464 The Structure Theorem for Abelian Groups 471 Application to Linear Operators 476 Free Modules over Polynomial Rings 482 EXERCISES 483 Fieids east nm Examples of Fields 492 Algebraic and Transcendental Elements 493 The Degree of a Field Extension 496 Constructions with Ruler and Compass 500 Symbolic Adjunction of Roots 506 Finite Fields 509 Function Fields 515 Transcendental Extensions 525 Algebraically Closed Fields 527 EXERCISES 530 Galois Theory ce» ama The Main Theorem of Galois Theory 537 Cubic Equations 543 Symmetric Functions 547 Primitive Elements 552 Proof of the Main Theorem 556 Quartic Equations 560 Kummer Extensions 565 Cyclotomic Extensions 567 Quintic Equations 570 EXERCISES 575 450 492 537 Preface Important though the general concepts and propositions may be with which the modern and industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless E am convinced that the special problems in all their complexity constitute the stock and core of mathematics, and that to master their difficulties requires on the whole the harder labor. Herman Weyl This book began about 20 years ago in the form of supplementary notes for my alge- bra classes. 1 wanted to discuss some concrete topics such as symmetry, linear groups, and quadratic number fields in more detail than the text provided, and to shift the emphasis in group theory from permutation groups to matrix groups. Lat- tices, another recurring theme, appeared spontaneously. My hope was that the con- crete material would interest the students and that it would make the abstractions more understandable, in short, that they could get farther by learning both at the same time. This worked pretty well. K took me quite a while to decide what 1 wanted to put in, but E gradually handed out more notes and eventually began teach- ing from them without another text. This method produced a book which is, I think, somewhat different from existing ones. However, the problems I encountered while fitting the parts together caused me many headaches, so I can't recommend starting this way. The main novel feature of the book is its increased emphasis on special topics. They tended to expand each time the sections were rewritten, because I noticed over the years that, with concrete mathematics in contrast to abstract concepts, students often prefer more to less. As a result, the ones mentioned above have become major parts of the book. There are also several unusual short subjects, such as the Todd- Coxeter algorithm and the simplicity of PSL, di xiv Preface In writing the book, 1 tried to follow these principles: 1, The main examples should precede the abstract definitions. 2. The book is not intended for a “service course,” so technical points should be presented only if they are needed in the book, 3. All topics discussed should be important for the average mathematician. Though these principles may sound like motherhood and the flag, I found it useful to have them enunciated, and to keep in mind that “Do it the way you were taught” isn't one of them. They are, of course, violated here and there. The table of contents gives a good idea of the subject matter, except that a first glance may lead you to believe that the book contains all of the standard material in a beginning algebra course, and more. Looking more closely, you will find that things have been pared down here and there to make space for the special topics. I used the above principles as a guide. Thus having the main examples in hand before proceeding to the abstract material allowed some abstractions to be treated more concisely. I was also able to shorten a few discussions by deferring them until the students have already overcome their inherent conceptual difficulties. The discussion of Peanos axioms in Chapter 10, for example, has been cut to two pages. Though the treatment given there is very incomplete, my experience is that it suffices to give the students the flavor of the axiomatic development of integer arithmetic. A more extensive discussion would be required if it were placed earlier in the book, and the time required for this wouldn't be well spent. Sometimes the exercise of deferring material showed that it could be deferred forever-—that it was not essential. This happened with dual spaces and multilinear algebra, for example, which wound up on the ficor as à consequence of the second principle. With a few concepts, such as the minimal polynomial, I ended vp believing that their main purpose in introductory al- gebra books has been to provide a convenient source Of exercises. The chapters are organized foilowing the order in which I usually teach a course, with linear algebra, group theory, and geometry making vp the first semester. Rings are first introduced in Chapter 10, though that chapter is logically independent of many earlier ones. J use this unusual arrangement because I want to emphasize the connections of algebra with geometry at the start, and because, over- all, the material in the first chapters is the most important for people in other fields. The drawback is that arithmetic is given short shrift. This is made up for in the later chapters, which have a strong arithmetic slant. Geometry is brought back from time to time in these later chapters, in the guise of lattices, symmetry, and algebraic ge- ometry. Michael Artin December 1990 xvi A Note for the Teacher tant in the beginning for the students who come to the course without a clear idea of what constitutes a proof. Chapter 1, matrix operations, isn't as exciting as some of the later ones, so it should be covered fairly quickly. 1 begin with it because I want to emphasize the general linear group at the start, instead of following the more customary practice of basing examples on the symmetric group. The reason for this decision is Principle 3 of the preface: The general linear group is more important. Here are some suggestions for Chapter 2: 1. Treat the abstract material with a light touch. You can have another go at it in Chapters 5 and 6. 2. For examples, concentrate on matrix groups. Mention permutation groups only in passing. Because of their inherent notational difficulties, examples from symme- try such as the dihedral groups are best deferred to Chapter 5. 3. Don't spend too much time on arithmetic. Its natural place in this book is Chap- ters 10 and 11. 4. Deemphasize the quotient group construction. Quotient groups present a pedagogica] problem. While their construction is concep- tually difficult, the quotient is readily presented as the image of a homomorphism in most elementary examples, and so it does not require an abstract definition. Modular arithmetic is about the only convincing example for which this is not the case. And since the integers modulo n form a ring, modular arithmetic isn't the ideal motivat- ing example for quotients of groups. The first serious use of quotient groups comes when generators and relations are discussed in Chapter 6, and 1 deferred the treat- ment of quotients to that point in early drafts of the book. But fearing the outrage of the algebra community 1 ended up moving it to Chapter 2. Anyhow, if you don't plan to discuss generators and relations for groups in your course, then you can defer an in-depth treatment of quotients to Chapter 10, ring theory, where they play a central role, and where modular arithmetic becomes a prime motivating example. In Chapter 3, vector spaces, T've tried to set up the computations with bases in such a way that the students won't have trouble keeping the indices straight. I've probabty failed, but since the notation is used throughout the book, it may be advis- able to adopt it. The applications of linear operators to rotations and linear differential equa- tions in Chapter 4 should be discussed because they are used later on, but the temp- tation to give differential equations their due has to be resisted. This heresy will be forgiven because you are teaching an algebra course. There is a gradual rise in the level of sophistication which is assumed of the reader throughout the first chapters, and a jump which T've been unable to eliminate occurs in Chapter 5. Had it not been for this jump, I would have moved symmetry closer to the beginning of the book. Keep in mind that symmetry is a difficult con- cept. It is easy to get carried away by the material and to leave the students behind. A Note for the Teacher xvii Except for its first two sections, Chapter 6 contains optional material. The last section on the Todd-Coxeter algorithm isn't standard; it is included to justify the discussion of generators and relations, which is pretty useless without it. There is nothing unusual in the chapter on bilinear forms, Chapter 7. I haven't overcome the main problem with this material, that there are too many variations on the same theme, but have tried to keep the discussion short by concentrating on the teal and complex cases, In the chapter on linear groups, Chapter 8, plan to spend time on the geometry of SU>. My students complained every year about this chapter until I expanded the sections on SU», after which they began asking for supplementary reading, wanting to leam more. Many of our students are not familiar with the concepts from topol- ogy when they take the course, and so these concepts reguire a light touch. But I've found that the problems caused by the students” lack of familiarity can be managed. Indeed, this is a good place for them to get an idea of what a manifold is. Unfortu- nately, I don't know a really satisfactory reference for further reading. Chapter 9 on group representations is optional. 1 resisted including this topic for a number of years, on the grounds that it is too hard. But students often request it, and I kept asking myself: Jf the chemists can teach it, why can't we? Eventually the internal logic of the book won out and group representations went in. As a divi- dend, hermitian forms got an application. The unusual topic in Chapter 11 is the arithmetic of quadratic number fields. You may find the discussion too long for a general algebra course. With this possibil- ity in mind, Pve arranged the material so that the end of Section 8, ideal factoriza- tion, is a natural stopping point. Tt seems to me that one should at least mention the most important examples of fields in a beginning algebra course, so I put a discussion of function fields into Chapter 13. There is always the question of whether or not Galois theory should be pre- sented in an undergraduate course. It doesn't have quite the universal applicability of most of the subjects in the book. But since Galois theory is a natural culmination of the discussion of symmetry, it belongs here as an optional topic. I usually spend at least some time on Chapter 14. I considered grading the exercises for difficulty, but found that I couldn't do it consistently. So I've only gone so far as to mark some of the harder ones with an asterisk. I believe that there are enough challenging problems, but of course one al- ways needs more of the interesting, easier ones. Though I've taught algebra for many years, several aspects of this book are ex- perimental, and I would be very grateful for critical comments and suggestions from the people who use it. “One, two, three, five, four...” “No Daddy, it's one, two, three, four, five.” “Weil if 1 want to say one, iwo, three, five, four, why can't PP” “Tha” s not how it goes.” Chapter 1 Matrix Operations Crfttich mid alles dasjenige eine Grópe genennt, melches einer Vegmebrung vber citer Becmindeçuna fabio ift, uber ioga dich noch ervas Gingufesen oder danor isegnelines Life, Leonhard Euler Matrices play a central role in this book. They form an important part of the theory, and many concrete examples are based on them. Therefore it is essential to develop facility in matrix manipulation. Since matrices pervade much of mathematics, the techniques needed here are sure to be useful elsewhere. The concepts which require practice to handle are matrix multiplication and determinants. 1. THE BASIC OPERATIONS Let m,n be positive integers. An m x n matrix is a collection of mn numbers ar- ranged in a rectangular array: n columns au ct am 1) m rows Gm tt Gm 2 For example, | 3] isa 2X 3 matrix. 135 The numbers in a matrix are called the matrix entries and are denoted by aj, where í, j are indices (integers) with | = i Gibi. Section 1 The Basic Operations 5 Each of these expressions is a shorthand notation for the sum (1.7) which defines the product matrix. Our two most important notations for handling sets of numbers are the X or sum notation as used above and matrix notation. The Z notation is actually the more versatile of the two, but because matrices are much more compact we will use them whenever possible. One of our tasks in later chapters will be to translate complicated mathematical structures into matrix notation in order to be able to work with them conveniently. Various idenfities are satisfied by the matrix operations, such as the distriburive laws (1.10) A(B+B)=AB+AB, and (A+A)B=AB+A'B and the associative law (1.11) (aBjc = A(BC). These laws hold whenever the matrices involved have suitable sizes, so that the products are defined. For the associative law, for example, the sizes should be A=tXm,B=mxXnand,C=nXp, for some É,m,n,p. Since the two products (1.11) are equal, the parentheses are not required, and we will denote them by ABC. The triple product ABC is then an € X p matrix. For example, the two ways of com- puting the product 1 20 ame = [au oa] a 01 are 20 me = [1 ; à| : | -f: y and ao = [iz n=[; E: Scalar multiplication is compatible with matrix multiplication in the obvious sense: (1.12) c(AB) = (cA)B = A(cB). The proofs of these identities are straightforward and not very interesting. Tn contrast, the commutative law does not hold for matrix multiplication; that is, (1.13) AB * BA, usually. In fect, if 4 is an € X m matrix and B is an m X € matrix, so that AB and BA are both defined, then 48 is € X £ while BA is m X m. Even if both matrices are square, say m Xm, the two products tend to be different. For instance, lo ollo 1) = [5 ol-sste lo Jo o)=[o o) 
