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STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 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 reprint-permission @ams.org. 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. c 2017 by the American Mathematical Society. All rights 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 222120191817 Contents Preface........................... vii Introduction........................ 1 §0.1. Notation..................... 3 §0.2. Zorn’sLemma.................. 5 Chapter 1. Integers and Integers mod n .......... 7 Chapter2.Groups..................... 13 §2.1. Groups, subgroups, and cosets . 13 §2.2. Group homomorphisms and factor groups . 25 §2.3. Groupactions.................. 32 §2.4. Symmetric and alternating groups . 36 §2.5. p-groups..................... 41 §2.6. Sylowsubgroups................. 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.Finiteabeliangroups .............. 66 Chapter3.Rings...................... 73 §3.1. Rings,subrings,andideals............ 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. Dualspace.................... 139 §4.4. Determinants .................. 142 §4.5. Eigenvalues and eigenvectors, triangulation and diagonalization . 150 §4.6. Minimal polynomials of a linear transformation andprimarydecomposition ........... 155 §4.7. T -cyclic subspaces and T -annihilators ...... 161 §4.8. Projectionmaps................. 164 §4.9. Cyclic decomposition and rational and Jordan canonicalforms................. 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. Splitting fields, normal field extensions, and Galois groups...................... 208 §5.6. Separability and repeated roots . 216 Contents v §5.7. Finitefields................... 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 SuggestionsforFurtherReading .............. 257 Bibliography........................ 259 IndexofNotation ..................... 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 chal- lenging “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 se- quence 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 the Second Isomorphism Theorem, etc. References for omitted proofs are provided except when they can be found in any text. Some of the problems given here appear as theorems or problems in many texts. They are included if they give results worth knowing or 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. I would also like to thank Skip Garibaldi for much valuable feedback and for suggesting some problems. Bibliography When available, Mathematical Reviews reference numbers are in- dicated at the end of each bibliographic entry as MR∗∗∗∗∗∗∗.See www.ams.org/mathscinet. 1. Artin, Michael, Algebra, second ed., Prentice Hall, Boston, MA, 2011. 2. Atiyah, Michael F. and Macdonald, Ian G., Introduction to commuta- tive algebra, Addison-Wesley Publishing Co., Reading, Mass.-London- Don Mills, Ont., 1969. MR0242802 3. Borevich, Zenon I. and Shafarevich, Igor R., Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. MR0195803 4. Cox, David A., Galois theory, second ed., Pure and Applied Math- ematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2012. MR2919975 5. Dummit, David S. and Foote, Richard M., Abstract algebra,thirded., John Wiley & Sons, Inc., Hoboken, NJ, 2004. MR2286236 6. Ebbinghaus, Heinz-Dieter; Hermes, Hans; Hirzebruch, Friedrich; Koecher, Max; Mainzer, Klaus; Neukirch, J¨urgen; Prestel, Alexander; and Remmert, Reinhold, Numbers, Graduate Texts in Mathematics, vol. 123, Springer-Verlag, New York, 1990, Readings in Mathematics. MR1066206 7. Hall, Jr., Marshall, The theory of groups, Chelsea Publishing Co., New York, 1976, Reprint of the 1968 edition. MR0414669 8. Hoffman, Kenneth and Kunze, Ray, Linear algebra, Second edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR0276251 259 260 Bibliography 9. Hungerford, Thomas W., Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980, Reprint
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