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Contemporary Abstract Algebra PDF

654 Pages·2020·28.378 MB·English
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Contemporary Abstract Algebra Textbooks in Mathematics Series editors: Al Boggess, Kenneth H. Rosen Mathematical Modeling in the Age of the Pandemic William P. Fox Linear Algebra James R. Kirkwood, Bessie H. Kirkwood Real Analysis With Proof Strategies Daniel W. Cunningham Train Your Brain Challenging Yet Elementary Mathematics Bogumil Kaminski, Pawel Pralat Contemporary Abstract Algebra, Tenth Edition Joseph A. Gallian Geometry and Its Applications Walter J. Meyer Linear Algebra What You Need to Know Hugo J. Woerdeman Introduction to Real Analysis, 3rd Edition Manfred Stoll Discovering Dynamical Systems Through Experiment and Inquiry Thomas LoFaro, Jeff Ford Functional Linear Algebra Hannah Robbins Introduction to Financial Mathematics With Computer Applications Donald R. Chambers, Qin Lu Linear Algebra An Inquiry-based Approach Jeff Suzuki Contemporary Abstract Algebra TENTH EDITION Joseph A. Gallian University of Minnesota Duluth Cover image by John Stembridge 2-dimensional representation of E8 Lie group First edition published 1986 D. C. Heath and Company 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN 10th edition © 2021 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and pub- lisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or here- after invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copy- right.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact HYPERLINK “mailto:[email protected][email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data ISBN: 978-0-367-65178-7 (hbk) ISBN: 978-1-003-14233-1 (ebk) Typeset in Computer Modern font by KnowledgeWorks Global Ltd. To Char, Krissy, and Joey Contents Notations xiii Preface xvii 0 Preliminaries 1 Properties of Integers 1 | Modular Arithmetic 5 | Complex Numbers 10 | Mathematical Induction 12 | Equivalence Relations 15|Functions (Mappings) 18 Exercises 21 1 Introduction to Groups 27 Symmetries of a Square 27|The Dihedral Groups 30 Biography of Niels Abel 38 2 Groups 39 Definition and Examples of Groups 39|Elementary Properties of Groups 47|Historical Note 50 Exercises 52 3 Finite Groups; Subgroups 59 Terminology and Notation 59|Subgroup Tests 61|Examples of Subgroups 64 Exercises 69 4 Cyclic Groups 77 Properties of Cyclic Groups 77|Classification of Subgroups of Cyclic Groups 83 Biography of James Joseph Sylvester 96 5 Permutation Groups 97 Definitions and Notation 97|Cycle Notation 100|Properties of Permutations 103|A Check-Digit Scheme Based on D 115 5 vii viii Contents Biography of Augustin Cauchy 125 Biography of Alan Turing 126 6 Isomorphisms 127 Motivation 127 | Definition and Examples 127 | Properties of Isomorphisms131|Automorphisms134|Cayley’sTheorem138 Exercises 141 Biography of Arthur Cayley 147 7 Cosets and Lagrange’s Theorem 149 Properties of Cosets 149|Lagrange’s Theorem and Consequences 153|An Application of Cosets to Permutation Groups 158 | The Rotation Group of a Cube and a Soccer Ball 160|An Application of Cosets to the Rubik’s Cube 163 Exercises 163 Biography of Joseph Lagrange 170 8 External Direct Products 171 Definition and Examples 171 | Properties of External Direct Products 173|The Group of Units Modulo n as an External Direct Product 176|Applications 178 Exercises 184 Biography of Leonard Adleman 191 9 Normal Subgroups and Factor Groups 193 Normal Subgroups 193 | Factor Groups 196 | Applications of Factor Groups 200|Internal Direct Products 203 Exercises 209 Biography of Évariste Galois 204 10 Group Homomorphisms 219 DefinitionandExamples219|PropertiesofHomomorphisms221 |The First Isomorphism Theorem 225 Exercises 232 Biography of Camille Jordan 239 11 Fundamental Theorem of Finite Abelian Groups 241 The Fundamental Theorem 241|The Isomorphism Classes of Abelian Groups 242|Proof of the Fundamental Theorem 246 Exercises 249 Contents ix 12 Introduction to Rings 255 Motivation and Definition 255 | Examples of Rings 256 | Properties of Rings 257|Subrings 259 Exercises 261 Biography of I. N. Herstein 266 13 Integral Domains 267 Definition and Examples 267|Fields 268|Characteristic of a Ring 271 Exercises 273 14 Ideals and Factor Rings 279 Ideals 279 | Factor Rings 280 | Prime Ideals and Maximal Ideals 284 Exercises 286 Biography of Richard Dedekind 293 Biography of Emmy Noether 294 15 Ring Homomorphisms 295 Definition and Examples 295|Properties of Ring Homomorphisms 298|The Field of Quotients 301 Exercises 303 16 Polynomial Rings 311 Notation and Terminology 311|The Division Algorithm and Consequences 314 Exercises 319 17 Factorization of Polynomials 325 Reducibility Tests 325|Irreducibility Tests 328|Unique Fac- torization in Z[x] 334|Weird Dice: An Application of Unique Factorization 335 Exercises 338 Biography of Serge Lang 342 18 Divisibility in Integral Domains 343 Irreducibles, Primes 343 | Historical Discussion of Fermat’s Last Theorem 346 | Unique Factorization Domains 350 | Euclidean Domains 353 Exercises 356 Biography of Sophie Germain 361

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