ABSTRACT ALGEBRA a study guide for ...

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ABSTRACTALGEBRA:
ASTUDYGUIDE
FORBEGINNERS
JohnA.Beachy
NorthernIllinoisUniversity
2012
ii
Thisisasupplementto
AbstractAlgebra,ThirdEdition
byJohnA.BeachyandWilliamD.Blair
ISBN1–57766–434–4,Copyright2005
WavelandPress,Inc.
4180ILRoute83,Suite101
LongGrove,Illinois60047
(847)634-0081
www.waveland.com
c
2000,2006,2012byJohnA.Beachy
Permissionisgrantedtocopythisdocumentinelectronicform,ortoprintitforpersonal
use,undertheseconditions:
itmustbereproducedinwhole;
itmustnotbemodifiedinanyway;
itmustnotbeusedaspartofanotherpublication.
FormattedJanuary6,2012,atwhichtimetheoriginalwasavailableat:
http://www.math.niu.edu/
beachy/abstractalgebra/
Contents
PREFACE
vi
1INTEGERS 1
1.1Divisors...................................... 1
1.2Primes....................................... 3
1.3Congruences.................................... 5
1.4IntegersModulon ................................ 8
Reviewproblems.................................... 10
2FUNCTIONS 13
2.1Functions..................................... 13
2.2EquivalenceRelations.............................. 16
2.3Permutations................................... 19
Reviewproblems.................................... 21
3GROUPS 23
3.1DefinitionofaGroup............................... 24
3.2Subgroups..................................... 28
3.3ConstructingExamples ............................. 32
3.4Isomorphisms................................... 35
3.5CyclicGroups................................... 38
3.6PermutationGroups............................... 40
3.7Homomorphisms................................. 42
3.8Cosets,NormalSubgroups,andFactorGroups................ 44
Reviewproblems.................................... 47
4POLYNOMIALS 49
4.1Fields;RootsofPolynomials........................... 49
4.2Factors....................................... 52
4.3ExistenceofRoots................................ 54
4.4PolynomialsoverZ,Q,R,andC........................ 55
Reviewproblems.................................... 57
iii
iv
CONTENTS
5COMMUTATIVERINGS 59
5.1Commutativerings;IntegralDomains..................... 59
5.2RingHomomorphisms.............................. 61
5.3IdealsandFactorRings............................. 63
5.4QuotientFields.................................. 64
Reviewproblems.................................... 65
6FIELDS 67
Reviewproblems.................................... 67
SOLUTIONS
68
1Integers 69
1.1Divisors...................................... 69
1.2Primes....................................... 74
1.3Congruences.................................... 78
1.4IntegersModulon ................................ 82
Reviewproblems.................................... 86
2Functions 89
2.1Functions..................................... 89
2.2EquivalenceRelations.............................. 94
2.3Permutations................................... 98
Reviewproblems....................................100
3Groups 103
3.1DefinitionofaGroup...............................103
3.2Subgroups.....................................109
3.3ConstructingExamples .............................114
3.4Isomorphisms...................................119
3.5CyclicGroups...................................125
3.6PermutationGroups...............................128
3.7Homomorphisms.................................130
3.8Cosets,NormalSubgroups,andFactorGroups................133
Reviewproblems....................................137
4Polynomials 143
4.1Fields;RootsofPolynomials...........................143
4.2Factors.......................................145
4.3ExistenceofRoots................................149
4.4PolynomialsoverZ,Q,R,andC........................151
Reviewproblems....................................153
CONTENTS
v
5CommutativeRings 157
5.1Commutativerings;IntegralDomains.....................157
5.2RingHomomorphisms..............................160
5.3IdealsandFactorRings.............................162
5.4QuotientFields..................................164
Reviewproblems....................................165
6Fields
169
BIBLIOGRAPHY
171
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