Complete Solutions Manual to Accompany
; ; ; ;
,ContemporaryAbstractAlgebra ; ;
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;c
opyright;herein;may;be;reproduced,;transmitted,;stored,;or;used Cengage;Learning
;in;any;form;or;by;any;means;graphic,;electronic,;or;mechanical, 200;First;Stamford;Place,;4th;Floor;Stamford
;including;but;not;limited;to;photocopying,;recording,;scanning,; ,;CT;06902
digitizing,;taping,;Web;distribution,;information;networks,;or;info USA
rmation;storage;and;retrieval;systems,;except;as;permitted;und
er;Section;107;or;108;of;the;1976;United;States;Copyright;Act,; Cengage;Learning;is;a;leading;provider;of;customized;learning;s
without;the;prior;written;permission;of;the;publisher;except;as;m olutions;with;office;locations;around;the;globe,;including;Singap
ay;be;permitted;by;the;license;terms;below. ore,;the;United;Kingdom,;Australia,;Mexico,;Brazil,;and;Japan.;L
ocate;your;local;office;at:;www.cengage.com/global.
Cengage;Learning;products;are;represented;in;Canada;by;Nels
For;product;information;and;technology;assistance,;contact;us;at on;Education,;Ltd.
Cengage;Learning;Customer;&;Sales;Support,
;1-800-354-9706. To;learn;more;about;Cengage;Learning;Solutions,;visit
www.cengage.com.
For;permission;to;use;material;from;this;text;or;product,;submit;all;req
uests;online;at;www.cengage.com/permissions
Purchase;any;of;our;products;at;your;local;college;store;or;at;our
Further;permissions;questions;can;be;emailed;to
. ;preferred;online;store;www.cengagebrain.com.
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;co
nnection;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;fr
om;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;Supple
ment;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;th
ey
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;fo
rm.;We;ask;that;you;take;reasonable;steps;to;protect;the;Supplement;from;unauthorized;use,;reproduction,;or;distribution.;Your;use;of;the;Supplement;indicate
s;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;Cenga
ge;Learning;and/or;its;licensors.;The;Supplement;is;furnished;by;Cengage;Learning;on;an;“as;is”;basis;without;any;warranties,;express;or;implied.;This;Agree
ment;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;teachi
ng;tool.
, 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
; ; ; ;
,ContemporaryAbstractAlgebra ; ;
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;c
opyright;herein;may;be;reproduced,;transmitted,;stored,;or;used Cengage;Learning
;in;any;form;or;by;any;means;graphic,;electronic,;or;mechanical, 200;First;Stamford;Place,;4th;Floor;Stamford
;including;but;not;limited;to;photocopying,;recording,;scanning,; ,;CT;06902
digitizing,;taping,;Web;distribution,;information;networks,;or;info USA
rmation;storage;and;retrieval;systems,;except;as;permitted;und
er;Section;107;or;108;of;the;1976;United;States;Copyright;Act,; Cengage;Learning;is;a;leading;provider;of;customized;learning;s
without;the;prior;written;permission;of;the;publisher;except;as;m olutions;with;office;locations;around;the;globe,;including;Singap
ay;be;permitted;by;the;license;terms;below. ore,;the;United;Kingdom,;Australia,;Mexico,;Brazil,;and;Japan.;L
ocate;your;local;office;at:;www.cengage.com/global.
Cengage;Learning;products;are;represented;in;Canada;by;Nels
For;product;information;and;technology;assistance,;contact;us;at on;Education,;Ltd.
Cengage;Learning;Customer;&;Sales;Support,
;1-800-354-9706. To;learn;more;about;Cengage;Learning;Solutions,;visit
www.cengage.com.
For;permission;to;use;material;from;this;text;or;product,;submit;all;req
uests;online;at;www.cengage.com/permissions
Purchase;any;of;our;products;at;your;local;college;store;or;at;our
Further;permissions;questions;can;be;emailed;to
. ;preferred;online;store;www.cengagebrain.com.
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;co
nnection;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;fr
om;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;Supple
ment;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;th
ey
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;fo
rm.;We;ask;that;you;take;reasonable;steps;to;protect;the;Supplement;from;unauthorized;use,;reproduction,;or;distribution.;Your;use;of;the;Supplement;indicate
s;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;Cenga
ge;Learning;and/or;its;licensors.;The;Supplement;is;furnished;by;Cengage;Learning;on;an;“as;is”;basis;without;any;warranties,;express;or;implied.;This;Agree
ment;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;teachi
ng;tool.
, 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
