Algebra: abstract and concrete

580 Pages·2012·5.9 MB·English

ALGEBRA
ABSTRACT CONCRETE
AND
EDITION 2.5
FREDERICK M. GOODMAN
SemiSimple Press
Iowa City, IA
LastrevisedonJune7,2012.
LibraryofCongressCataloging-in-PublicationData
Goodman,FrederickM.
Algebra:abstractandconcrete /FrederickM.Goodman—ed.2.5
p. cm.
Includesindex.
ISBN 978-0-9799142-0-1
1.Algebra. I.Title
QA155.R642006
512–dc21
(cid:13)c 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.
ISBN 978-0-9799142-0-1
Contents
Preface vii
A Note to the Reader ix
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 44
1.9. Counting 56
1.10. Groups 69
1.11. RingsandFields 75
1.12. AnApplicationtoCryptography 79
Chapter 2. Basic Theory of Groups 84
2.1. FirstResults 84
2.2. SubgroupsandCyclicGroups 93
2.3. TheDihedralGroups 105
2.4. HomomorphismsandIsomorphisms 110
2.5. CosetsandLagrange’sTheorem 120
2.6. EquivalenceRelationsandSetPartitions 126
2.7. QuotientGroupsandHomomorphismTheorems 133
Chapter 3. Products of Groups 147
3.1. DirectProducts 147
3.2. SemidirectProducts 155
3.3. VectorSpaces 158
3.4. Thedualofavectorspaceandmatrices 173
3.5. LinearalgebraoverZ 185
3.6. Finitelygeneratedabeliangroups 194
Chapter 4. Symmetries of Polyhedra 211
4.1. RotationsofRegularPolyhedra 211
iii
iv CONTENTS
4.2. RotationsoftheDodecahedronandIcosahedron 220
4.3. WhataboutReflections? 224
4.4. LinearIsometries 229
4.5. TheFullSymmetryGroupandChirality 234
Chapter 5. Actions of Groups 237
5.1. GroupActionsonSets 237
5.2. GroupActions—CountingOrbits 244
5.3. SymmetriesofGroups 247
5.4. GroupActionsandGroupStructure 250
5.5. Application: TransitiveSubgroupsofS 259
5
5.6. AdditionalExercisesforChapter5 261
Chapter 6. Rings 264
6.1. ARecollectionofRings 264
6.2. HomomorphismsandIdeals 270
6.3. QuotientRings 282
6.4. IntegralDomains 289
6.5. EuclideanDomains,PrincipalIdeal
Domains,andUniqueFactorization 294
6.6. UniqueFactorizationDomains 303
6.7. NoetherianRings 310
6.8. IrreducibilityCriteria 313
Chapter 7. Field Extensions – First Look 317
7.1. ABriefHistory 317
7.2. SolvingtheCubicEquation 318
7.3. AdjoiningAlgebraicElementstoaField 322
7.4. SplittingFieldofaCubicPolynomial 328
7.5. SplittingFieldsofPolynomialsinCŒx(cid:141) 337
Chapter 8. Modules 345
8.1. Theideaofamodule 345
8.2. Homomorphismsandquotientmodules 353
8.3. Multilinearmapsanddeterminants 357
8.4. FinitelygeneratedModulesoveraPID,partI 369
8.5. FinitelygeneratedModulesoveraPID,partII. 379
8.6. Rationalcanonicalform 392
8.7. JordanCanonicalForm 407
Chapter 9. Field Extensions – Second Look 420
9.1. FiniteandAlgebraicExtensions 420
9.2. SplittingFields 422
9.3. TheDerivativeandMultipleRoots 425
9.4. SplittingFieldsandAutomorphisms 427
CONTENTS v
9.5. TheGaloisCorrespondence 434
9.6. SymmetricFunctions 440
9.7. TheGeneralEquationofDegreen 448
9.8. QuarticPolynomials 456
9.9. GaloisGroupsofHigherDegreePolynomials 462
Chapter 10. Solvability 468
10.1. CompositionSeriesandSolvableGroups 468
10.2. CommutatorsandSolvability 470
10.3. SimplicityoftheAlternatingGroups 472
10.4. CyclotomicPolynomials 475
10.5. TheEquationxn(cid:0)b D 0 478
10.6. SolvabilitybyRadicals 480
10.7. RadicalExtensions 483
Chapter 11. Isometry Groups 487
11.1. MoreonIsometriesofEuclideanSpace 487
11.2. Euler’sTheorem 494
11.3. FiniteRotationGroups 497
11.4. Crystals 501
Appendix A. Almost Enough about Logic 520
A.1. Statements 520
A.2. LogicalConnectives 521
A.3. Quantifiers 525
A.4. Deductions 527
Appendix B. Almost Enough about Sets 528
B.1. FamiliesofSets;UnionsandIntersections 532
B.2. FiniteandInfiniteSets 533
Appendix C. Induction 535
C.1. ProofbyInduction 535
C.2. DefinitionsbyInduction 536
C.3. MultipleInduction 537
Appendix D. Complex Numbers 540
Appendix E. Review of Linear Algebra 542
E.1. LinearalgebrainKn 542
E.2. BasesandDimension 547
E.3. InnerProductandOrthonormalBases 551
Appendix F. Models of Regular Polyhedra 553
Appendix G. Suggestions for Further Study 561
Index 563

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.
Supplements.
I maintain a World Wide Web site with electronic supplements to the
text, at http://www.math.uiowa.edu/~goodman. Materials available
atthissitemayinclude
(cid:15) Color versions of graphics from the text and manipulable three-
dimensionalgraphics
(cid:15) Programsforalgebraiccomputations
(cid:15) Errata
vii
viii PREFACE
I would be grateful for any comments on the text, reports of errors, and
suggestions for improvements. I am currently distributing this text elec-
tronically, and this means that I can provide frequent updates and correc-
tions. Please write if you would like a better text next semester! I thank
thosestudentsandinstructorswhohavewrittenmeinthepast.
FrederickM.Goodman
[email protected]
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.
ix
x ANOTETOTHEREADER
“ThenIknew,”hesaid,“iftherewerefliesinthisholetheyhad
to come from the hole above that’s in the sunlight where there’s
enoughheattomakethemhatch.
“Afterthat,Ishouldhaveseenthemdeadinthewater. SinceI
couldn’tseethemdeadinthewater,Iknewtheyhadtobeatleast
sixorseveninchesunderthewaterwhereIcouldn’tseethem. So
that’swhereIfished.”
He leaned against the rock with his hands behind his head to
make the rock soft. “Wade out there and try George’s No. 2,” he
said,pointingattheflyhehadgivenme. 1
In mathematical practice the typical experience is to be faced by a
problem whose solution is an mystery: The fish won’t bite. Even if you
haveatoolboxfullofmethodsandrules,theproblemdoesn’tcomelabeled
with the applicable method, and the rules don’t seem to fit. There is no
otherwaybuttothinkthingsthroughforyourself.
Thepurposeofthiscourseistointroduceyoutothepracticeofmathe-
matics;tohelpyoulearntothinkthingsthroughforyourself;toteachyou
tosee“somethingnoticeablewhichmakesyouseesomethingyouweren’t
noticingwhichmakesyouseesomethingthatisn’tevenvisible.”Andthen
toexplainaccuratelywhatyouhaveunderstood.
Notincidentally,thecourseaimstoshowyousomealgebraicandgeo-
metricideasthatareinterestingandimportantandworththinkingabout.
It’snotatalleasytolearntoworkthingsoutforyourself,andit’snot
atalleasytoexplainclearlywhatyouhaveworkedout. Theseartshaveto
belearnedbythoughtfulpractice.
You must have patience, or learn patience, and you must have time.
Youcan’tlearnthesethingswithoutgettingfrustrated,andyoucan’tlearn
them in a hurry. If you can get someone else to explain how to do the
problems,youwilllearnsomething,butnotpatience,andnotpersistence,
andnotvision. Sorelyonyourselfasfaraspossible.
But rely on your teacher as well. Your teacher will give you hints,
suggestions,andinsightsthatcanhelpyouseeforyourself. Abookalone
cannotdothis,becauseitcannotlistentoyouandrespond.
Iwishyousuccess,andIhopeyouwillsomedayfishinwatersnotyet
dreamed of. Meanwhile, I have arranged a tour of some well known but
interestingstreams.
1FromNormanMaclean,ARiverRunsThroughIt,UniversityofChicagoPress,1976.
Reprintedbypermission.