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STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board SatyanL. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary00A07, 12-01, 13-01, 15-01, 20-01. For additional informationand updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth,AdrianR.,1947– Title: Problemsinabstractalgebra/A.R.Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical referencesandindex. 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(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 222120191817 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1 §0.1. Notation . . . . . . . . . . . . . . . . . . . . . 3 §0.2. Zorn’s Lemma . . . . . . . . . . . . . . . . . . 5 Chapter 1. Integers and Integers mod n . . . . . . . . . . 7 Chapter 2. Groups . . . . . . . . . . . . . . . . . . . . . 13 §2.1. Groups, subgroups, and cosets . . . . . . . . . . 13 §2.2. Group homomorphisms and factor groups . . . . 25 §2.3. Group actions . . . . . . . . . . . . . . . . . . 32 §2.4. Symmetric and alternating groups . . . . . . . . 36 §2.5. p-groups . . . . . . . . . . . . . . . . . . . . . 41 §2.6. Sylow subgroups . . . . . . . . . . . . . . . . . 43 §2.7. Semidirect products of groups . . . . . . . . . . 44 §2.8. Free groups and groups by generators and relations 53 §2.9. Nilpotent, solvable, and simple groups . . . . . . 58 §2.10. Finite abelian groups . . . . . . . . . . . . . . 66 Chapter 3. Rings . . . . . . . . . . . . . . . . . . . . . . 73 §3.1. Rings, subrings, and ideals . . . . . . . . . . . . 73 iii iv Contents §3.2. Factor rings and ring homomorphisms . . . . . . 89 §3.3. Polynomial rings and evaluation maps . . . . . . 97 §3.4. Integral domains, quotient fields . . . . . . . . . 100 §3.5. Maximal ideals and prime ideals . . . . . . . . . 103 §3.6. Divisibility and principal ideal domains . . . . . . 107 §3.7. Unique factorization domains . . . . . . . . . . 115 Chapter 4. Linear Algebra and Canonical Forms of Linear Transformations . . . . . . . . . . . . . . . . 125 §4.1. Vector spaces and linear dependence . . . . . . . 125 §4.2. Linear transformations and matrices . . . . . . . 132 §4.3. Dual space . . . . . . . . . . . . . . . . . . . . 139 §4.4. Determinants . . . . . . . . . . . . . . . . . . 142 §4.5. Eigenvalues and eigenvectors, triangulation and diagonalization . . . . . . . . . . . . . . . . . 150 §4.6. Minimal polynomials of a linear transformation and primary decomposition . . . . . . . . . . . 155 §4.7. T-cyclic subspaces and T-annihilators . . . . . . 161 §4.8. Projection maps . . . . . . . . . . . . . . . . . 164 §4.9. Cyclic decomposition and rational and Jordan canonical forms . . . . . . . . . . . . . . . . . 167 §4.10. The exponential of a matrix . . . . . . . . . . . 177 §4.11. Symmetric and orthogonal matrices over R . . . . 180 §4.12. Group theory problems using linear algebra . . . 187 Chapter 5. Fields and Galois Theory . . . . . . . . . . . . 191 §5.1. Algebraic elements and algebraic field extensions . 192 §5.2. Constructibility by compass and straightedge . . . 199 §5.3. Transcendental extensions . . . . . . . . . . . . 202 §5.4. Criteria for irreducibility of polynomials . . . . . 205 §5.5. Splittingfields, normalfieldextensions, andGalois groups . . . . . . . . . . . . . . . . . . . . . . 208 §5.6. Separability and repeated roots . . . . . . . . . 216 Contents v §5.7. Finite fields . . . . . . . . . . . . . . . . . . . 223 §5.8. Galois field extensions . . . . . . . . . . . . . . 226 §5.9. Cyclotomic polynomials and cyclotomic extensions 234 §5.10. Radical extensions, norms, and traces . . . . . . 244 §5.11. Solvability by radicals . . . . . . . . . . . . . . 253 Suggestions for Further Reading . . . . . . . . . . . . . . 257 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 259 Index of Notation . . . . . . . . . . . . . . . . . . . . . 261 Subject and Terminology Index . . . . . . . . . . . . . . . 267 Preface It is a truism that, for most students, solving problems is a vital part of learning a mathematical subject well. Furthermore, I think students learn the most from challenging problems that demand se- rious thought and help develop a deeper understanding of important ideas. When teaching abstract algebra, I found it frustrating that most textbooks did not have enough interesting or really demanding problems. This led me to provide regular supplementary problem handouts. The handouts usually included a few particularly challenging “optional problems,” which the students were free to work on or not, but they would receive some extra credit for turning in correct solutions. My problem handouts were the primary source for the problems in this book. They were used in teaching Math 100, University of California, San Diego’s yearlong honors level course sequence in abstract algebra, and for the first term of Math 200, the graduate abstract algebra sequence. I hope this problem book will be a useful resource for students learning abstract algebra and for professors wanting to go beyond their textbook’s problems. To make this book somewhat more self-contained and indepen- dent of any particular text, I have included definitions of most con- cepts and statements of key theorems (including a few short proofs). This will mitigate the problem that texts do not completely agree on some definitions; likewise, there are different names and versions of vii viii Preface some theorems. For example, what is here called the Fundamental Homomorphism Theorem many authors call the First Isomorphism Theorem; so what is here the First Isomorphism Theorem they call theSecondIsomorphismTheorem,etc. Referencesforomittedproofs are provided except when they can be found in any text. Someoftheproblemsgivenhereappearastheoremsorproblems inmanytexts. Theyareincludediftheygiveresultsworthknowingor if they are building blocks for other problems. Many of the problems have multiple parts; this makes it possible to develop a topic more thoroughly than what can be done in just a few sentences. Multipart problems can also provide paths to more difficult results. I would like to thank Richard Elman for helpful suggestions and references. IwouldalsoliketothankSkipGaribaldiformuchvaluable feedback and for suggesting some problems. Introduction Thisbookisacollectionofproblemsinabstractalgebraforstrong advanced undergraduates or beginning graduate students in mathematics. Some of the problems will be challenging even for very tal- ented students. These problems can be used by students taking an abstract algebra course who want more challenge or some interesting enrichment to their course. They can also be used by more experi- encedstudentsforreviewortosolidifytheirknowledgeofthesubject. Professors teaching algebra courses may use this book as a source to supplement the problems from their textbook. The assumed background for those undertaking these problems includes familiarity with the basic set-theoretic language of mathematics and the ability to write rigorous mathematical proofs. For Chapters 4 and 5, rudimentary knowledge of linear algebra is also needed. Studentsshouldprobablybetakingorhavetakenanabstract algebra course or be reading an abstract algebra text concurrently. No solutions are provided for the problems given here (though therearemanyhints). Theguidingphilosophyforthisisthatreaders who do not succeed witha first effort at a difficult problem can often progressandlearnmorebygoingbacktoitatalatertime. Solutions inthebackofthebookoffertoomuchtemptationtogiveupworking on a problem too soon. 1