Download Algebra: Abstract and Concrete, edition 2.6.

588 Pages·2015·5.92 MB·English

ALGEBRA
ABSTRACT CONCRETE
AND
EDITION 2.6
FREDERICK M. GOODMAN
SemiSimple Press
Iowa City, IA
LastrevisedonMay1,2015.
Algebra:abstractandconcrete /FrederickM.Goodman—ed.2.6
ISBN 978-0-9799142-1-8
(cid:13)c 2014,2006,2003,1998byFrederickM.Goodman
SemiSimplePress
IowaCity,IA
Theauthorreservesallrightstothisworknotexplicitlygranted,includingtherighttocopy,reproduceand
distributetheworkinanyform,printedorelectronic,byanymeans,inwholeorinpart.However,individualreaders,
classesorstudygroupsmaycopy,storeandprintthework,inwholeorinpart,fortheirpersonaluse.Anycopyof
thiswork,oranypartofit,mustincludethetitlepagewiththeauthor’snameandthiscopyrightnotice.
Nouseorreproductionofthisworkforcommercialpurposesispermittedwithoutthewrittenpermissionof
theauthor.Thisworkmaynotbeadaptedoralteredwithouttheauthor’swrittenconsent.
ThefirstandsecondeditionsofthisworkwerepublishedbyPrentice-Hall. Thecurrent
versionofthistextisavailablefromhttp://www.math.uiowa.edu/~goodman.
ISBN 978-0-9799142-1-8
Contents
Preface vii
The Price of this Book ix
A Note to the Reader x
Chapter 1. Algebraic Themes 1
1.1. WhatIsSymmetry? 1
1.2. SymmetriesoftheRectangleandtheSquare 3
1.3. MultiplicationTables 7
1.4. SymmetriesandMatrices 11
1.5. Permutations 16
1.6. DivisibilityintheIntegers 24
1.7. ModularArithmetic 37
1.8. Polynomials 45
1.9. Counting 56
1.10. Groups 69
1.11. RingsandFields 75
1.12. AnApplicationtoCryptography 80
Chapter 2. Basic Theory of Groups 85
2.1. FirstResults 85
2.2. SubgroupsandCyclicGroups 94
2.3. TheDihedralGroups 106
2.4. HomomorphismsandIsomorphisms 111
2.5. CosetsandLagrange’sTheorem 121
2.6. EquivalenceRelationsandSetPartitions 127
2.7. QuotientGroupsandHomomorphismTheorems 134
Chapter 3. Products of Groups 149
3.1. DirectProducts 149
3.2. SemidirectProducts 160
3.3. VectorSpaces 163
3.4. Thedualofavectorspaceandmatrices 178
3.5. LinearalgebraoverZ 190
3.6. Finitelygeneratedabeliangroups 199
iii
iv CONTENTS
Chapter 4. Symmetries of Polyhedra 216
4.1. RotationsofRegularPolyhedra 216
4.2. RotationsoftheDodecahedronandIcosahedron 225
4.3. WhataboutReflections? 229
4.4. LinearIsometries 234
4.5. TheFullSymmetryGroupandChirality 239
Chapter 5. Actions of Groups 242
5.1. GroupActionsonSets 242
5.2. GroupActions—CountingOrbits 249
5.3. SymmetriesofGroups 252
5.4. GroupActionsandGroupStructure 255
5.5. Application: TransitiveSubgroupsofS 264
5
5.6. AdditionalExercisesforChapter5 266
Chapter 6. Rings 269
6.1. ARecollectionofRings 269
6.2. HomomorphismsandIdeals 275
6.3. QuotientRings 288
6.4. IntegralDomains 295
6.5. EuclideanDomains,PrincipalIdeal
Domains,andUniqueFactorization 300
6.6. UniqueFactorizationDomains 309
6.7. NoetherianRings 316
6.8. IrreducibilityCriteria 319
Chapter 7. Field Extensions – First Look 322
7.1. ABriefHistory 322
7.2. SolvingtheCubicEquation 323
7.3. AdjoiningAlgebraicElementstoaField 327
7.4. SplittingFieldofaCubicPolynomial 334
7.5. SplittingFieldsofPolynomialsinCŒx(cid:141) 342
Chapter 8. Modules 350
8.1. Theideaofamodule 350
8.2. Homomorphismsandquotientmodules 358
8.3. Multilinearmapsanddeterminants 362
8.4. FinitelygeneratedModulesoveraPID,partI 374
8.5. FinitelygeneratedModulesoveraPID,partII. 385
8.6. Rationalcanonicalform 398
8.7. JordanCanonicalForm 413
Chapter 9. Field Extensions – Second Look 426
9.1. FiniteandAlgebraicExtensions 426
9.2. SplittingFields 428
CONTENTS v
9.3. TheDerivativeandMultipleRoots 431
9.4. SplittingFieldsandAutomorphisms 433
9.5. TheGaloisCorrespondence 441
9.6. SymmetricFunctions 446
9.7. TheGeneralEquationofDegreen 453
9.8. QuarticPolynomials 461
9.9. GaloisGroupsofHigherDegreePolynomials 468
Chapter 10. Solvability 473
10.1. CompositionSeriesandSolvableGroups 473
10.2. CommutatorsandSolvability 475
10.3. SimplicityoftheAlternatingGroups 477
10.4. CyclotomicPolynomials 480
10.5. TheEquationxn(cid:0)b D 0 483
10.6. SolvabilitybyRadicals 485
10.7. RadicalExtensions 488
Chapter 11. Isometry Groups 492
11.1. MoreonIsometriesofEuclideanSpace 492
11.2. Euler’sTheorem 499
11.3. FiniteRotationGroups 502
11.4. Crystals 506
Appendix A. Almost Enough about Logic 525
A.1. Statements 525
A.2. LogicalConnectives 526
A.3. Quantifiers 530
A.4. Deductions 532
Appendix B. Almost Enough about Sets 533
B.1. FamiliesofSets;UnionsandIntersections 537
B.2. FiniteandInfiniteSets 538
Appendix C. Induction 540
C.1. ProofbyInduction 540
C.2. DefinitionsbyInduction 541
C.3. MultipleInduction 542
Appendix D. Complex Numbers 545
Appendix E. Review of Linear Algebra 547
E.1. LinearalgebrainKn 547
E.2. BasesandDimension 552
E.3. InnerProductandOrthonormalBases 556
Appendix F. Models of Regular Polyhedra 558
vi CONTENTS
Appendix G. Suggestions for Further Study 566
Index 568
Preface
Thistextprovidesathoroughintroductionto“modern”or“abstract”alge-
bra at a level suitable for upper-level undergraduates and beginning grad-
uatestudents.
Thebookaddressestheconventionaltopics: groups,rings,fields,and
linear algebra, with symmetry as a unifying theme. This subject matter is
centralandubiquitousinmodernmathematicsandinapplicationsranging
fromquantumphysicstodigitalcommunications.
The most important goal of this book is to engage students in the ac-
tivepracticeofmathematics. Studentsaregiventheopportunitytopartici-
pateandinvestigate,startingonthefirstpage. Exercisesareplentiful,and
workingexercisesshouldbetheheartofthecourse.
The required background for using this text is a standard first course
in linear algebra. I have included a brief summary of linear algebra in an
appendixtohelpstudentsreview. Ihavealsoprovidedappendicesonsets,
logic, mathematical induction, and complex numbers. It might also be
useful to recommend a short supplementary text on set theory, logic, and
proofs to be used as a reference and aid; several such texts are currently
available.
Acknowledgements.
The first and second editions of this text were published by Prentice
Hall. I would like to thank George Lobell, the staff at Prentice Hall, and
reviewersofthepreviouseditionsfortheirhelpandadvice.
Thankstomanyreadersforsuggestionsandcorrections. Thanksespe-
ciallytoWenJiaLiuforcompilingalonglistofcorrections.
Currentversionandsupplements.
Thecurrentversionofthistextisavailablefrom
http://www.math.uiowa.edu/~goodman.
Some supplementary materials are available at the same site, including
manipulable three-dimensional graphics and programs for algebraic com-
putations.
I would be grateful for any comments on the text, reports of errors,
and suggestions for improvements. I am currently distributing this text
vii
viii PREFACE
electronically,andthismeansthatIcanprovidefrequentupdatesandcor-
rections. Pleasewriteifyouwouldlikeabettertextnextsemester! Ithank
thosestudentsandinstructorswhohavewrittenmeinthepast.
FrederickM.Goodman
[email protected]
The Price of this Book
If you have the time and opportunity to study abstract algebra, it is likely
thatyouarenothungry,coldandsick.
Thisbookisbeingofferedfreeofchargeforyouruse. Inexchange,ifyou
makeserioususeofthisbook, pleasemakeacontributiontorelievingthe
miseryoftheworld.
For example, you could make a financial contribution to an organization
suchasUnicef,DoctorswithoutBorders,PartnersinHealth,orOxfam,or
to an equivalent organization in your country. Or you could find a way to
volunteeryourtimeandknowledgeinstead.
ix
A Note to the Reader
Iwouldliketoshowyouapassagefromoneofmyfavoritebooks,ARiver
Runs Through It, by Norman Maclean. The narrator Norman is fishing
withhisbrotherPaulonamountainriverneartheirhomeinMontana. The
brothers have been fishing a “hole” blessed with sunlight and a hatch of
yellowstoneflies,onwhichthefisharevigorouslyfeeding. Theydescend
to the next hole downstream, where the fish will not bite. After a while
Paul,whoisfishingtheoppositesideoftheriver,makessomeadjustment
to his equipment and begins to haul in one fish after another. Norman
watchesinfrustrationandadmiration,untilPaulwadesovertohissideof
therivertohandhimafly:
He gave me a pat on the back and one of George’s No. 2 Yel-
low Hackles with a feather wing. He said, “They are feeding on
drownedyellowstoneflies.”
Iaskedhim,“Howdidyouthinkthatout?”
He thought back on what had happened like a reporter. He
started to answer, shook his head when he found he was wrong,
and then started out again. “All there is to thinking,” he said, “is
seeingsomethingnoticeablewhichmakesyouseesomethingyou
weren’t noticing which makes you see something that isn’t even
visible.”
I said to my brother, “Give me a cigarette and say what you
mean.”
“Well,” he said, “the first thing I noticed about this hole was
thatmybrotherwasn’tcatchingany. There’snothingmorenotice-
abletoafishermanthanthathispartnerisn’tcatchingany.
“This made me see that I hadn’t seen any stone flies flying
aroundthishole.”
Then he asked me, “What’s more obvious on earth
thansunshineandshadow,butuntilIreallysawthattherewereno
stone flies hatching here I didn’t notice that the upper hole where
they were hatching was mostly in sunshine and this hole was in
shadow.”
I was thirsty to start with, and the cigarette made my mouth
drier,soIflippedthecigaretteintothewater.
x