Cornerstones Series Editors Charles L. Epstein, University of Pennsylvania, Philadelphia Steven G. Krantz, University of Washington, St. Louis Advisory Board Anthony W. Knapp, State University of New York at Stony Brook, Emeritus Anthony W. Knapp Basic Algebra Along with a companion volume Advanced Algebra Birkhauser¨ Boston • Basel • Berlin Anthony W. Knapp 81 Upper Sheep Pasture Road East Setauket, NY 11733-1729 U.S.A. e-mail to: [email protected] http://www.math.sunysb.edu/˜ aknapp/books/b-alg.html Cover design by Mary Burgess. Mathematics Subject Classicification (2000): 15-01, 20-02, 13-01, 12-01, 16-01, 08-01, 18A05, 68P30 Library of Congress Control Number: 2006932456 ISBN-10 0-8176-3248-4 eISBN-10 0-8176-4529-2 ISBN-13 978-0-8176-3248-9 eISBN-13 978-0-8176-4529-8 Advanced Algebra ISBN 0-8176-4522-5 Basic Algebra and Advanced Algebra (Set) ISBN 0-8176-4533-0 Printed on acid-free paper. c 2006 Anthony W. Knapp All rights reserved. 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The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 www.birkhauser.com (EB) To Susan and To My Algebra Teachers: Ralph Fox, John Fraleigh, Robert Gunning, John Kemeny, Bertram Kostant, Robert Langlands, Goro Shimura, Hale Trotter, Richard Williamson CONTENTS Contents of Advanced Algebra x List of Figures xi Preface xiii Dependence Among Chapters xvii Standard Notation xviii Guide for the Reader xix I. PRELIMINARIES ABOUT THE INTEGERS, POLYNOMIALS, AND MATRICES 1 1. Division and Euclidean Algorithms 1 2. Unique Factorization of Integers 4 3. Unique Factorization of Polynomials 9 4. Permutations and Their Signs 15 5. Row Reduction 19 6. Matrix Operations 24 7. Problems 30 II. VECTOR SPACES OVER Q, R, AND C 33 1. Spanning, Linear Independence, and Bases 33 2. Vector Spaces Defined by Matrices 38 3. Linear Maps 42 4. Dual Spaces 50 5. Quotients of Vector Spaces 54 6. Direct Sums and Direct Products of Vector Spaces 58 7. Determinants 65 8. Eigenvectors and Characteristic Polynomials 73 9. Bases in the Infinite-Dimensional Case 77 10. Problems 82 III. INNER-PRODUCT SPACES 88 1. Inner Products and Orthonormal Sets 88 2. Adjoints 98 3. Spectral Theorem 104 4. Problems 111 vii viii Contents IV. GROUPS AND GROUP ACTIONS 116 1. Groups and Subgroups 117 2. Quotient Spaces and Homomorphisms 128 3. Direct Products and Direct Sums 134 4. Rings and Fields 140 5. Polynomials and Vector Spaces 147 6. Group Actions and Examples 158 7. Semidirect Products 166 8. Simple Groups and Composition Series 170 9. Structure of Finitely Generated Abelian Groups 174 10. Sylow Theorems 183 11. Categories and Functors 188 12. Problems 198 V. THEORY OF A SINGLE LINEAR TRANSFORMATION 209 1. Introduction 209 2. Determinants over Commutative Rings with Identity 212 3. Characteristic and Minimal Polynomials 216 4. Projection Operators 224 5. Primary Decomposition 226 6. Jordan Canonical Form 229 7. Computations with Jordan Form 235 8. Problems 239 VI. MULTILINEAR ALGEBRA 245 1. Bilinear Forms and Matrices 246 2. Symmetric Bilinear Forms 250 3. Alternating Bilinear Forms 253 4. Hermitian Forms 255 5. Groups Leaving a Bilinear Form Invariant 257 6. Tensor Product of Two Vector Spaces 260 7. Tensor Algebra 274 8. Symmetric Algebra 280 9. Exterior Algebra 288 10. Problems 292 VII. ADVANCED GROUP THEORY 303 1. Free Groups 303 2. Subgroups of Free Groups 314 3. Free Products 319 4. Group Representations 326 Contents ix VII. ADVANCED GROUP THEORY (Continued) 5. Burnside’s Theorem 342 6. Extensions of Groups 344 7. Problems 357 VIII. COMMUTATIVE RINGS AND THEIR MODULES 367 1. Examples of Rings and Modules 367 2. Integral Domains and Fields of Fractions 378 3. Prime and Maximal Ideals 381 4. Unique Factorization 384 5. Gauss’s Lemma 390 6. Finitely Generated Modules 396 7. Orientation for Algebraic Number Theory and Algebraic Geometry 408 8. Noetherian Rings and the Hilbert Basis Theorem 414 9. Integral Closure 417 10. Localization and Local Rings 425 11. Dedekind Domains 434 12. Problems 439 IX. FIELDS AND GALOIS THEORY 448 1. Algebraic Elements 449 2. Construction of Field Extensions 453 3. Finite Fields 457 4. Algebraic Closure 460 5. Geometric Constructions by Straightedge and Compass 464 6. Separable Extensions 469 7. Normal Extensions 476 8. Fundamental Theorem of Galois Theory 479 9. Application to Constructibility of Regular Polygons 483 10. Application to Proving the Fundamental Theorem of Algebra 486 11. Application to Unsolvability of Polynomial Equations with Nonsolvable Galois Group 488 12. Construction of Regular Polygons 493 13. Solution of Certain Polynomial Equations with Solvable Galois Group 501 14. Proof That π Is Transcendental 510 15. Norm and Trace 514 16. Splitting of Prime Ideals in Extensions 521 17. Two Tools for Computing Galois Groups 527 18. Problems 534 x Contents X. MODULES OVER NONCOMMUTATIVE RINGS 544 1. Simple and Semisimple Modules 544 2. Composition Series 551 3. Chain Conditions 556 4. Hom and End for Modules 558 5. Tensor Product for Modules 565 6. Exact Sequences 574 7. Problems 579 APPENDIX 583 A1. Sets and Functions 583 A2. Equivalence Relations 589 A3. Real Numbers 591 A4. Complex Numbers 594 A5. Partial Orderings and Zorn’s Lemma 595 A6. Cardinality 599 Hints for Solutions of Problems 603 Selected References 697 Indexof Notation 699 Index 703 CONTENTS OF ADVANCEDALGEBRA I. Transition to Modern Number Theory II. Wedderburn–Artin Ring Theory III. Brauer Group IV. Homological Algebra V. Three Theorems in Algebraic Number Theory VI. Reinterpretation with Adeles and Ideles VII. Infinite Field Extensions VIII. Background for Algebraic Geometry IX. The Number Theory of Algebraic Curves X. Methods of Algebraic Geometry LIST OF FIGURES 2.1. The vector space of lines v + U in R2 parallel to a given line U through the origin 55 2.2. Factorization of linear maps via a quotient of vector spaces 56 2.3. Three 1-dimensional vector subspaces of R2 such that each pair has intersection 062 2.4. Universal mapping property of a direct product of vector spaces 64 2.5. Universal mapping property of a direct sum of vector spaces 65 3.1. Geometric interpretation of the parallelogram law 91 3.2. Resolution of a vector into a parallel component and an orthogonal component 93 4.1. Factorization of homomorphisms of groups via the quotient of a group by a normal subgroup 132 4.2. Universal mapping property of an external direct product of groups 136 4.3. Universal mapping property of a direct product of groups 136 4.4. Universal mapping property of an external direct sum of abelian groups 138 4.5. Universal mapping property of a direct sum of abelian groups 139 4.6. Factorization of homomorphisms of rings via the quotient of a ring by an ideal 146 4.7. Substitution homomorphism for polynomials in one indeterminate 150 4.8. Substitution homomorphism for polynomials in n indeterminates 156 4.9. A square diagram 193 4.10. Diagrams obtained by applying a covariant functor and a contravariant functor 193 4.11. Universal mapping property of a product in a category 195 4.12. Universal mapping property of a coproduct in a category 197 5.1. Example of a nilpotent matrix in Jordan form 231 5.2. Powers of the nilpotent matrix in Figure 5.1 232 6.1. Universal mapping property of a tensor product 261 6.2. Diagrams for uniqueness of a tensor product 261 xi xii List of Figures 6.3. Commutative diagram of a natural transformation {TX } 265 6.4. Commutative diagram of a triple tensor product 274 6.5. University mapping property of a tensor algebra 279 7.1. Universal mapping property of a free group 305 7.2. Universal mapping property of a free product 320 7.3. An intertwining operator for two representations 330 7.4. Equivalent group extensions 349 8.1. Universal mapping property of the integral group ring of G 371 8.2. Universal mapping property of a free left R module 374 8.3. Factorization of R homomorphisms via a quotient of R modules 376 8.4. Universal mapping property of the group algebra RG 378 8.5. Universal mapping property of the field of fractions of R 380 8.6. Real points of the curve y2 = (x − 1)x(x + 1) 409 8.7. Universal mapping property of the localization of R at S 428 9.1. Closure of positive constructible x coordinates under multiplication and division 465 9.2. Closure of positive constructible x coordinates under square roots 466 9.3. Construction of a regular pentagon 496 9.4. Construction of a regular 17-gon 500 10.1. Universal mapping property of a tensor product of a right R module and a left R module 566 PREFACE Basic Algebra and its companion volume Advanced Algebra systematically de- velop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. These two books together aim to give the reader a global view of algebra, its use, and its role in mathematics as a whole.