Advanced Modern Algebra Third Edition, Part 1 Joseph J. Rotman Graduate Studies in Mathematics Volume 165 American Mathematical Society Advanced Modern Algebra Third Edition, Part 1 https://doi.org/10.1090//gsm/165 Advanced Modern Algebra Third Edition, Part 1 Joseph J. Rotman Graduate Studies in Mathematics Volume 165 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani The 2002 edition of this book was previously published by Pearson Education, Inc. 2010 Mathematics Subject Classification. Primary 12-01, 13-01, 14-01, 15-01, 16-01, 18-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-165 Library of Congress Cataloging-in-Publication Data Rotman, Joseph J., 1934– Advanced modern algebra / Joseph J. Rotman. – Third edition. volumes cm. – (Graduate studies in mathematics ; volume 165) Includes bibliographical references and index. ISBN 978-1-4704-1554-9 (alk. paper : pt. 1) 1. Algebra. I. Title. QA154.3.R68 2015 512–dc23 2015019659 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. Third edition c 2015 by the American Mathematical Society. All rights reserved. Second edition c 2010 by the American Mathematical Society. All rights reserved. First edition c 2002 by the American Mathematical Society. All right reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 To my wife Marganit and our two wonderful kids Danny and Ella, whom I love very much Contents Preface to Third Edition: Part 1 xi Acknowledgments xiv Part A. Course I Chapter A-1. Classical Formulas 3 Cubics 4 Quartics 6 Chapter A-2. Classical Number Theory 9 Divisibility 9 Euclidean Algorithms 16 Congruence 19 Chapter A-3. Commutative Rings 29 Polynomials 41 Homomorphisms 47 Quotient Rings 55 From Arithmetic to Polynomials 62 Maximal Ideals and Prime Ideals 74 Finite Fields 83 Irreducibility 89 Euclidean Rings and Principal Ideal Domains 97 Unique Factorization Domains 104 Chapter A-4. Groups 115 Permutations 116 vii viii Contents Even and Odd 123 Groups 127 Lagrange’s Theorem 139 Homomorphisms 150 Quotient Groups 159 Simple Groups 173 Chapter A-5. Galois Theory 179 Insolvability of the Quintic 179 Classical Formulas and Solvability by Radicals 187 Translation into Group Theory 190 Fundamental Theorem of Galois Theory 200 Calculations of Galois Groups 223 Chapter A-6. Appendix: Set Theory 235 Equivalence Relations 243 Chapter A-7. Appendix: Linear Algebra 247 Vector Spaces 247 Linear Transformations and Matrices 259 Part B. Course II Chapter B-1. Modules 273 Noncommutative Rings 273 Chain Conditions on Rings 282 Left and Right Modules 288 Chain Conditions on Modules 300 Exact Sequences 305 Chapter B-2. Zorn’s Lemma 313 Zorn, Choice, and Well-Ordering 313 Zorn and Linear Algebra 319 Zorn and Free Abelian Groups 323 Semisimple Modules and Rings 334 Algebraic Closure 339 Transcendence 345 L¨uroth’s Theorem 353 Chapter B-3. Advanced Linear Algebra 359 Torsion and Torsion-free 359 Basis Theorem 362 Contents ix Fundamental Theorem 371 Elementary Divisors 371 Invariant Factors 374 From Abelian Groups to Modules 378 Rational Canonical Forms 383 Eigenvalues 388 Jordan Canonical Forms 395 Smith Normal Forms 402 Inner Product Spaces 417 Orthogonal and Symplectic Groups 429 Hermitian Forms and Unitary Groups 436 Chapter B-4. Categories of Modules 441 Categories 441 Functors 461 Galois Theory for Infinite Extensions 475 Free and Projective Modules 481 Injective Modules 492 Divisible Abelian Groups 501 Tensor Products 509 Adjoint Isomorphisms 522 Flat Modules 529 Chapter B-5. Multilinear Algebra 543 Algebras and Graded Algebras 543 Tensor Algebra 552 Exterior Algebra 561 Grassmann Algebras 566 Exterior Algebra and Differential Forms 573 Determinants 575 Chapter B-6. Commutative Algebra II 591 Old-Fashioned Algebraic Geometry 591 Affine Varieties and Ideals 593 Nullstellensatz 599 Nullstellensatz Redux 604 Irreducible Varieties 614 Affine Morphisms 623 Algorithms in k[x1,...,xn] 628 Monomial Orders 629 x Contents Division Algorithm 636 Gr¨obner Bases 639 Chapter B-7. Appendix: Categorical Limits 651 Inverse Limits 651 Direct Limits 657 Directed Index Sets 659 Adjoint Functors 666 Chapter B-8. Appendix: Topological Spaces 673 Topological Groups 678 Bibliography 681 Special Notation 687 Index 693 Preface to Third Edition: Part 1 Algebra is used by virtually all mathematicians, be they analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. Nowa- days, everyone agrees that some knowledge of linear algebra, group theory, and commutative algebra is necessary, and these topics are introduced in undergrad- uate courses. Since there are many versions of undergraduate algebra courses, I will often review definitions, examples, and theorems, sometimes sketching proofs and sometimes giving more details.1 Part 1 of this third edition can be used as a text for the first year of graduate algebra, but it is much more than that. It and the forthcoming Part 2 can also serve more advanced graduate students wishing to learn topics on their own. While not reaching the frontiers, the books provide a sense of the successes and methods arising in an area. In addition, they comprise a reference containing many of the standard theorems and definitions that users of algebra need to know. Thus, these books are not merely an appetizer, they are a hearty meal as well. When I was a student, Birkhoff–Mac Lane, A Survey of Modern Algebra [8], was the text for my first algebra course, and van der Waerden, Modern Algebra [118], was the text for my second course. Both are excellent books (I have called this book Advanced Modern Algebra in homage to them), but times have changed since their first publication: Birkhoff and Mac Lane’s book appeared in 1941; van der Waerden’s book appeared in 1930. There are today major directions that either did not exist 75 years ago, or were not then recognized as being so important, or were not so well developed. These new areas involve algebraic geometry, category 1It is most convenient for me, when reviewing earlier material, to refer to my own text FCAA: A First Course in Abstract Algebra,3rded.[94], as well as to LMA, the book of A. Cuoco and myself [23], Learning Modern Algebra from Early Attempts to Prove Fermat’s Last Theorem. xi xii Preface to Third Edition: Part 1 theory,2 computer science, homological algebra, and representation theory. Each generation should survey algebra to make it serve the present time. The passage from the second edition to this one involves some significant changes, the major change being organizational. This can be seen at once, for the elephantine 1000 page edition is now divided into two volumes. This change is not merely a result of the previous book being too large; instead, it reflects the structure of beginning graduate level algebra courses at the University of Illinois at Urbana–Champaign. This first volume consists of two basic courses: Course I (Galois theory) followed by Course II (module theory). These two courses serve as joint prerequisites for the forthcoming Part 2, which will present more advanced topics in ring theory, group theory, algebraic number theory, homological algebra, representation theory, and algebraic geometry. In addition to the change in format, I have also rewritten much of the text. For example, noncommutative rings are treated earlier. Also, the section on alge- braic geometry introduces regular functions and rational functions. Two proofs of the Nullstellensatz (which describes the maximal ideals in k[x1,...,xn]whenk is an algebraically closed field) are given. The first proof, for k = C (which easily generalizes to uncountable k), is the same proof as in the previous edition. But the second proof I had written, which applies to countable algebraically closed fields as well, was my version of Kaplansky’s account [55] of proofs of Goldman and of Krull. I should have known better! Kaplansky was a master of exposition, and this edition follows his proof more closely. The reader should look at Kaplansky’s book, Selected Papers and Writings [58], to see wonderful mathematics beautifully expounded. I have given up my attempted spelling reform, and I now denote the ring of integers mod m by Zm instead of by Im. A star * before an exercise indicates that it will be cited elsewhere in the book, possibly in a proof. The first part of this volume is called Course I; it follows a syllabus for an actual course of lectures. If I were king, this course would be a transcript of my lectures.