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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.

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Jun 7, 2012 Algebra: abstract and concrete / Frederick M. Goodman— ed. 2.5 p. cm The most important goal of this book is to engage students in the ac-.
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