Complete Solutions Manual to Accompany
Contemporary Abstract Algebra
NINTH EDITION
Joseph Gallian
University of Minnesota Duluth
Prepared by
Joseph Gallian
University of Minnesota Duluth
Australia • Brazil • Mexico • Singapore • United Kingdom • United States
, © 2017 Cengage Learning ISBN-13: 978-13056579-84
ISBN-10: 0-130565798-5
ALL RIGHTS RESERVED. No part of this work covered by the
copyright herein may be reproduced, transmitted, stored, or Cengage Learning
used in any form or by any means graphic, electronic, or 200 First Stamford Place, 4th Floor
mechanical, including but not limited to photocopying, Stamford, CT 06902
recording, scanning, digitizing, taping, Web distribution, USA
information networks, or information storage and retrieval
systems, except as permitted under Section 107 or 108 of the Cengage Learning is a leading provider of customized learning
1976 United States Copyright Act, without the prior written solutions with office locations around the globe, including
permission of the publisher except as may be permitted by the Singapore, the United Kingdom, Australia, Mexico, Brazil, and
license terms below. Japan. Locate your local office at: www.cengage.com/global.
Cengage Learning products are represented in Canada by
Nelson Education, Ltd.
For product information and technology assistance, contact us at
Cengage Learning Customer & Sales Support, To learn more about Cengage Learning Solutions, visit
1-800-354-9706. www.cengage.com.
For permission to use material from this text or product, submit all
Purchase any of our products at your local college store or at
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READ IMPORTANT LICENSE INFORMATION
Dear Professor or Other Supplement Recipient:
Cengage Learning has provided you with this product (the “Supplement”) for your review and, to the extent that you adopt the associated textbook for use in
connection with your course (the “Course”), you and your students who purchase the textbook may use the Supplement as described below. Cengage
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stemming from unlimited distribution of Supplements.
Cengage Learning hereby grants you a nontransferable license to use the Supplement in connection with the Course, subject to the following conditions. The
Supplement is for your personal, noncommercial use only and may not be reproduced, posted electronically or distributed, except that portions of the
Supplement may be provided to your students IN PRINT FORM ONLY in connection with your instruction of the Course, so long as such students are
advised that they
may not copy or distribute any portion of the Supplement to any third party. You may not sell, license, auction, or otherwise redistribute the Supplement in any
form. We ask that you take reasonable steps to protect the Supplement from unauthorized use, reproduction, or distribution. Your use of the Supplement
indicates your acceptance of the conditions set forth in this Agreement. If you do not accept these conditions, you must return the Supplement unused within
30 days of receipt.
All rights (including without limitation, copyrights, patents, and trade secrets) in the Supplement are and will remain the sole and exclusive property of
Cengage Learning and/or its licensors. The Supplement is furnished by Cengage Learning on an “as is” basis without any warranties, express or implied.
This Agreement will be governed by and construed pursuant to the laws of the State of New York, without regard to such State’s conflict of law rules.
Thank you for your assistance in helping to safeguard the integrity of the content contained in this Supplement. We trust you find the Supplement a useful
teaching tool.
, i
CONTEMPORARY ABSTRACT ALGEBRA 9TH EDITION
INSTRUCTOR SOLUTION MANUAL
CONTENTS
Integers and Equivalence Relations
0 Preliminaries 1
Introduction to Groups 7
Groups
2 Groups 9
3 Finite Groups; Subgroups 13
1 4 Cyclic Groups 20
5 Permutation Groups 27
6 Isomorphisms 34
7 Cosets and Lagrange’s Theorem 40
8 External Direct Products 46
9 Normal Subgroups and Factor Groups 53
10 Group Homomorphisms 59
11 Fundamental Theorem of Finite Abelian Groups 65
12 Introduction to Rings 69
13 Integral Domains 74
14 Ideals and Factor Rings 80
15 Ring Homomorphisms 87
16 Polynomial Rings 94
17 Factorization of Polynomials 100
18 Divisibility in Integral Domains 105
, ii
Fields
19 Vector Spaces 110
20 Extension Fields 114
21 Algebraic Extensions 118
22 Finite Fields 123
23 Geometric Constructions 127
Special Topics
24 Sylow Theorems 129
25 Finite Simple Groups 135
26 Generators and Relations 140
27 Symmetry Groups 144
28 Frieze Groups and Crystallographic Groups 146
29 Symmetry and Counting 148
30 Cayley Digraphs of Groups 151
31 Introduction to Algebraic Coding Theory 154
32 An Introduction to Galois Theory 158
33 Cyclotomic Extensions 161
Contemporary Abstract Algebra
NINTH EDITION
Joseph Gallian
University of Minnesota Duluth
Prepared by
Joseph Gallian
University of Minnesota Duluth
Australia • Brazil • Mexico • Singapore • United Kingdom • United States
, © 2017 Cengage Learning ISBN-13: 978-13056579-84
ISBN-10: 0-130565798-5
ALL RIGHTS RESERVED. No part of this work covered by the
copyright herein may be reproduced, transmitted, stored, or Cengage Learning
used in any form or by any means graphic, electronic, or 200 First Stamford Place, 4th Floor
mechanical, including but not limited to photocopying, Stamford, CT 06902
recording, scanning, digitizing, taping, Web distribution, USA
information networks, or information storage and retrieval
systems, except as permitted under Section 107 or 108 of the Cengage Learning is a leading provider of customized learning
1976 United States Copyright Act, without the prior written solutions with office locations around the globe, including
permission of the publisher except as may be permitted by the Singapore, the United Kingdom, Australia, Mexico, Brazil, and
license terms below. Japan. Locate your local office at: www.cengage.com/global.
Cengage Learning products are represented in Canada by
Nelson Education, Ltd.
For product information and technology assistance, contact us at
Cengage Learning Customer & Sales Support, To learn more about Cengage Learning Solutions, visit
1-800-354-9706. www.cengage.com.
For permission to use material from this text or product, submit all
Purchase any of our products at your local college store or at
requests online at www.cengage.com/permissions
our preferred online store www.cengagebrain.com.
Further permissions questions can be emailed to
.
READ IMPORTANT LICENSE INFORMATION
Dear Professor or Other Supplement Recipient:
Cengage Learning has provided you with this product (the “Supplement”) for your review and, to the extent that you adopt the associated textbook for use in
connection with your course (the “Course”), you and your students who purchase the textbook may use the Supplement as described below. Cengage
Learning has established these use limitations in response to concerns raised by authors, professors, and other users regarding the pedagogical problems
stemming from unlimited distribution of Supplements.
Cengage Learning hereby grants you a nontransferable license to use the Supplement in connection with the Course, subject to the following conditions. The
Supplement is for your personal, noncommercial use only and may not be reproduced, posted electronically or distributed, except that portions of the
Supplement may be provided to your students IN PRINT FORM ONLY in connection with your instruction of the Course, so long as such students are
advised that they
may not copy or distribute any portion of the Supplement to any third party. You may not sell, license, auction, or otherwise redistribute the Supplement in any
form. We ask that you take reasonable steps to protect the Supplement from unauthorized use, reproduction, or distribution. Your use of the Supplement
indicates your acceptance of the conditions set forth in this Agreement. If you do not accept these conditions, you must return the Supplement unused within
30 days of receipt.
All rights (including without limitation, copyrights, patents, and trade secrets) in the Supplement are and will remain the sole and exclusive property of
Cengage Learning and/or its licensors. The Supplement is furnished by Cengage Learning on an “as is” basis without any warranties, express or implied.
This Agreement will be governed by and construed pursuant to the laws of the State of New York, without regard to such State’s conflict of law rules.
Thank you for your assistance in helping to safeguard the integrity of the content contained in this Supplement. We trust you find the Supplement a useful
teaching tool.
, i
CONTEMPORARY ABSTRACT ALGEBRA 9TH EDITION
INSTRUCTOR SOLUTION MANUAL
CONTENTS
Integers and Equivalence Relations
0 Preliminaries 1
Introduction to Groups 7
Groups
2 Groups 9
3 Finite Groups; Subgroups 13
1 4 Cyclic Groups 20
5 Permutation Groups 27
6 Isomorphisms 34
7 Cosets and Lagrange’s Theorem 40
8 External Direct Products 46
9 Normal Subgroups and Factor Groups 53
10 Group Homomorphisms 59
11 Fundamental Theorem of Finite Abelian Groups 65
12 Introduction to Rings 69
13 Integral Domains 74
14 Ideals and Factor Rings 80
15 Ring Homomorphisms 87
16 Polynomial Rings 94
17 Factorization of Polynomials 100
18 Divisibility in Integral Domains 105
, ii
Fields
19 Vector Spaces 110
20 Extension Fields 114
21 Algebraic Extensions 118
22 Finite Fields 123
23 Geometric Constructions 127
Special Topics
24 Sylow Theorems 129
25 Finite Simple Groups 135
26 Generators and Relations 140
27 Symmetry Groups 144
28 Frieze Groups and Crystallographic Groups 146
29 Symmetry and Counting 148
30 Cayley Digraphs of Groups 151
31 Introduction to Algebraic Coding Theory 154
32 An Introduction to Galois Theory 158
33 Cyclotomic Extensions 161
