A Graduate Course in Algebra: 1

445 Pages·2017·2.551 MB·English

by  Ioannis Farmakis, Martin Moskowitz

A GRADUATE
COURSE IN
ALGEBRA

10106_9789813142626_TP_v1.indd 1 31/5/17 2:48 PM
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A GRADUATE
COURSE IN
ALGEBRA
Volume 1
Ioannis Farmakis
Department of Mathematics, Brooklyn College
City University of New York, USA
Martin Moskowitz

Ph.D. Program in Mathematics, CUNY Graduate Center
City University of New York, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
10106_9789813142626_TP_v1.indd 2 31/5/17 2:48 PM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Farmakis, Ioannis. | Moskowitz, Martin A.
Title: A graduate course in algebra : in 2 volumes / by Ioannis Farmakis
(City University of New York, USA), Martin Moskowitz (City University of New York, USA).
Description: New Jersey : World Scientific, 2017– |
Includes bibliographical references and index.
Identifiers: LCCN 2017001101| ISBN 9789813142626 (hardcover : alk. paper : v. 1) |
ISBN 9789813142633 (pbk : alk. paper : v. 1) | 9789813142664 (hardcover : alk. paper : v. 2) |
ISBN 9789813142671 (pbk : alk. paper : v. 2) | ISBN 9789813142602 (set : alk. paper) |
ISBN 9789813142619 (pbk set : alk. paper)
Subjects: LCSH: Algebra--Textbooks. | Algebra--Study and teaching (Higher)
Classification: LCC QA154.3 .F37 2017 | DDC 512--dc23
LC record available at https://lccn.loc.gov/2017001101
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
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Printed in Singapore
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Contents
Preface and Acknowledgments xi
0 Introduction 1
0.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Cartesian Product . . . . . . . . . . . . . . . . . . . . . 3
0.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.4 Partially Ordered Sets . . . . . . . . . . . . . . . . . . . 6
0.4.1 The Principle of Induction . . . . . . . . . . . . . 6
0.4.2 Transfinite Induction . . . . . . . . . . . . . . . . 9
0.4.3 Permutations . . . . . . . . . . . . . . . . . . . . 9
0.5 The set (Z,+,×) . . . . . . . . . . . . . . . . . . . . . . 11
0.5.1 The Fundamental Theorem of Arithmetic . . . . 13
0.5.2 The Euler Formula and Riemann Zeta Function. 17
0.5.3 The Fermat Numbers . . . . . . . . . . . . . . . 21
0.5.4 Pythagorean Triples . . . . . . . . . . . . . . . . 23
1 Groups 27
1.1 The Concept of a Group . . . . . . . . . . . . . . . . . . 27
1.2 Examples of Groups . . . . . . . . . . . . . . . . . . . . 31
1.2.1 The Quaternion Group . . . . . . . . . . . . . . 33
1.2.2 The Dihedral Group . . . . . . . . . . . . . . . . 35
1.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.3.0.1 Exercises . . . . . . . . . . . . . . . . . 40
1.4 Quotient Groups . . . . . . . . . . . . . . . . . . . . . . 41
1.4.1 Cosets . . . . . . . . . . . . . . . . . . . . . . . . 41
v
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vi Contents
1.5 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . 44
1.5.1 Chinese Remainder Theorem . . . . . . . . . . . 48
1.5.2 Fermat’s Little Theorem . . . . . . . . . . . . . . 50
1.5.3 Wilson’s Theorem . . . . . . . . . . . . . . . . . 55
1.5.3.1 Exercises . . . . . . . . . . . . . . . . . 57
1.6 Automorphisms, Characteristic and Normal Subgroups . 58
1.7 The Center of a Group, Commutators . . . . . . . . . . 62
1.8 The Three Isomorphism Theorems . . . . . . . . . . . . 63
1.9 Groups of Low Order . . . . . . . . . . . . . . . . . . . . 66
1.9.0.1 Exercises . . . . . . . . . . . . . . . . . 70
1.10 Direct and Semi-direct Products . . . . . . . . . . . . . 71
1.10.1 Direct Products. . . . . . . . . . . . . . . . . . . 71
1.10.2 Semi-Direct Products . . . . . . . . . . . . . . . 75
1.11 Exact Sequences of Groups . . . . . . . . . . . . . . . . 79
1.12 Direct and Inverse Limits of Groups . . . . . . . . . . . 82
1.13 Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . 89
1.14 Some Features of Abelian Groups . . . . . . . . . . . . . 92
1.14.1 Torsion Groups . . . . . . . . . . . . . . . . . . . 92
1.14.2 Divisible and Injective Groups . . . . . . . . . . 96
1.14.3 Pru¨fer Groups . . . . . . . . . . . . . . . . . . . 100
1.14.4 Structure Theorem for Divisible Groups . . . . . 102
1.14.5 Maximal Subgroups . . . . . . . . . . . . . . . . 104
1.14.5.1 Exercises . . . . . . . . . . . . . . . . . 105
2 Further Topics in Group Theory 109
2.1 Composition Series . . . . . . . . . . . . . . . . . . . . . 109
2.2 Solvability, Nilpotency . . . . . . . . . . . . . . . . . . . 112
2.2.1 Solvable Groups . . . . . . . . . . . . . . . . . . 112
2.2.2 Nilpotent Groups . . . . . . . . . . . . . . . . . . 117
2.2.2.1 Exercises . . . . . . . . . . . . . . . . . 120
2.3 Group Actions . . . . . . . . . . . . . . . . . . . . . . . 121
2.3.1 Stabilizer and Orbit . . . . . . . . . . . . . . . . 124
2.3.2 Transitive Actions . . . . . . . . . . . . . . . . . 126
2.3.3 Some Examples of Transitive Actions . . . . . . 128
2.3.3.1 The Real Sphere Sn−1 . . . . . . . . . . 128
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Contents vii
2.3.3.2 The Complex Sphere S2n−1 . . . . . . . 129
2.3.3.3 The Quaternionic Sphere S4n−1 . . . . 130
2.3.3.4 The Poincar´e Upper Half-Plane . . . . 131
2.3.3.5 Real Projective Space RPn . . . . . . . 132
2.3.3.6 Complex Projective Space CPn . . . . . 133
2.3.3.7 The Grassmann and Flag Varieties . . . 135
2.4 PSL(n,k), A & Iwasawa’s Double Transitivity Theorem 138
n
2.5 Imprimitive Actions . . . . . . . . . . . . . . . . . . . . 145
2.6 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . 147
2.7 The Three Sylow Theorems . . . . . . . . . . . . . . . . 148
2.7.0.1 Exercises . . . . . . . . . . . . . . . . . 152
3 Vector Spaces 155
3.1 Generation, Basis and Dimension . . . . . . . . . . . . . 157
3.1.0.1 Exercises . . . . . . . . . . . . . . . . . 162
3.2 The First Isomorphism Theorem . . . . . . . . . . . . . 162
3.2.1 Systems of Linear Equations . . . . . . . . . . . 164
3.2.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . 166
3.3 Second and Third Isomorphism Theorems . . . . . . . . 168
3.4 Linear Transformations and Matrices . . . . . . . . . . . 170
3.4.1 Eigenvalues, Eigenvectors and Diagonalizability . 173
3.4.1.1 The Fibonacci Sequence . . . . . . . . . 175
3.4.2 Application to Matrix Differential Equations . . 178
3.4.2.1 Exercises . . . . . . . . . . . . . . . . . 181
3.5 The Dual Space . . . . . . . . . . . . . . . . . . . . . . . 181
3.5.1 Annihilators. . . . . . . . . . . . . . . . . . . . . 183

3.5.2 Systems of Linear Equations Revisited . . . . . . 184
3.5.3 The Adjoint of an Operator . . . . . . . . . . . . 186
3.6 Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . 187
3.7 Tensor Products . . . . . . . . . . . . . . . . . . . . . . 189
3.7.1 Tensor Products of Linear Operators . . . . . . . 194
3.8 Complexification of a Real Vector Space . . . . . . . . . 198
3.8.1 Complexifying with Tensor Products . . . . . . . 201
3.8.2 Real Forms and Complex Conjugation . . . . . . 205
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viii Contents
4 Inner Product Spaces 207
4.0.1 Gram-Schmidt Orthogonalization . . . . . . . . . 213
4.0.1.1 Legendre Polynomials . . . . . . . . . . 214
4.0.2 Bessel’s Inequality and Parseval’s Equation . . . 216
4.1 Subspaces and their Orthocomplements . . . . . . . . . 218
4.2 The Adjoint Operator . . . . . . . . . . . . . . . . . . . 220
4.3 Unitary and Orthogonal Operators . . . . . . . . . . . . 221
4.3.1 Eigenvalues of Orthogonal and Unitary Operators 223
4.4 Symmetric and Hermitian Operators . . . . . . . . . . . 223
4.4.1 Skew-Symmetric and Skew-Hermitian Operators 225
4.5 The Cayley Transform . . . . . . . . . . . . . . . . . . . 228
4.6 Normal Operators . . . . . . . . . . . . . . . . . . . . . 230
4.6.1 The Spectral Theorem . . . . . . . . . . . . . . . 231
4.7 Some Applications to Lie Groups . . . . . . . . . . . . . 238
4.7.1 Positive Definite Operators . . . . . . . . . . . . 238
4.7.1.1 Exercises . . . . . . . . . . . . . . . . . 242
4.7.2 The Topology on H+ and P+ . . . . . . . . . . . 243
4.7.3 The Polar Decomposition . . . . . . . . . . . . . 245
4.7.4 The Iwasawa Decomposition . . . . . . . . . . . 247
4.7.4.1 |det| and Volume in Rn. . . . . . . . . 250
4.7.5 The Bruhat Decomposition . . . . . . . . . . . . 252
4.8 Gramians . . . . . . . . . . . . . . . . . . . . . . . . . . 259
4.9 Schur’s Theorems and Eigenvalue Estimates . . . . . . . 263
4.10 The Geometry of the Conics . . . . . . . . . . . . . . . . 267
4.10.1 Polarization of Symmetric Bilinear Forms . . . . 267
4.10.2 Classification of Quadric Surfaces under Aff(V) . 268

5 Rings, Fields and Algebras 277
5.1 Preliminary Notions . . . . . . . . . . . . . . . . . . . . 277
5.2 Subrings and Ideals . . . . . . . . . . . . . . . . . . . . . 284
5.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . 285
5.3.1 The Three Isomorphism Theorems . . . . . . . . 286
5.3.2 The Characteristic of a Field . . . . . . . . . . . 287
5.4 Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . . 289
5.4.1 Prime Ideals . . . . . . . . . . . . . . . . . . . . 291
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Contents ix
5.5 Euclidean Rings . . . . . . . . . . . . . . . . . . . . . . 293
5.5.1 Vieta’s Formula and the Discriminant . . . . . . 297
5.5.2 The Chinese Remainder Theorem . . . . . . . . . 298
5.6 Unique Factorization . . . . . . . . . . . . . . . . . . . . 303
5.6.1 Fermat’s Two-Square Thm. & Gaussian Primes . 305
5.7 The Polynomial Ring . . . . . . . . . . . . . . . . . . . . 309
5.7.1 Gauss’ Lemma and Eisenstein’s Criterion . . . . 310
5.7.2 Cyclotomic Polynomials . . . . . . . . . . . . . . 313
5.7.3 The Formal Derivative . . . . . . . . . . . . . . . 316
5.7.4 The Fundamental Thm. of Symmetric
Polynomials . . . . . . . . . . . . . . . . . . . . . 320
5.7.5 There is No Such Thing as a Pattern . . . . . . . 324
5.7.6 Non-Negative Polynomials . . . . . . . . . . . . . 325
5.7.6.1 Exercises . . . . . . . . . . . . . . . . . 328
5.8 Finite Fields and Wedderburn’s Little Theorem . . . . . 330
5.8.1 Application to Projective Geometry . . . . . . . 333
5.9 k-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 335
5.9.1 Division Algebras . . . . . . . . . . . . . . . . . . 341
5.9.1.1 Exercises . . . . . . . . . . . . . . . . . 344
6 R-Modules 347
6.1 Generalities on R-Modules . . . . . . . . . . . . . . . . . 347
6.1.1 The Three Isomorphism Theorems for Modules . 350
6.1.2 Direct Products and Direct Sums . . . . . . . . . 351
6.2 Homological Algebra . . . . . . . . . . . . . . . . . . . . 353
6.2.1 Exact Sequences . . . . . . . . . . . . . . . . . . 354
6.2.2 Free Modules . . . . . . . . . . . . . . . . . . . . 362
6.2.3 The Tensor Product of R-modules . . . . . . . . 366
6.3 R-Modules vs. Vector Spaces . . . . . . . . . . . . . . . 373
6.4 Finitely Generated Modules over a Euclidean Ring . . . 375
6.4.0.1 Exercises . . . . . . . . . . . . . . . . . 379
6.5 Applications to Linear Transformations . . . . . . . . . 379
6.6 The Jordan Canonical Form and Jordan Decomposition 385
6.6.1 The Minimal Polynomial (continued...). . . . . . 389
6.6.2 Families of Commuting Operators . . . . . . . . 391
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x Contents
6.6.3 Additive & Multiplicative Jordan Decompositions 393
6.7 The Jordan-H¨older Theorem for R-Modules . . . . . . . 397
6.8 The Fitting Decomposition and Krull-Schmidt Theorem 400
6.9 Schur’s Lemma and Simple Modules . . . . . . . . . . . 403
Appendix 407
Appendix A: Pell’s Equation 407
Appendix B: The Kronecker Approximation Theorem 414
Appendix C: Some Groups of Automorphisms 416
Bibliography 419
Index 429