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Algebra. Abstract and concrete PDF

587 Pages·2015·4.866 MB·Russian
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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 symmetryas aunifying theme. Thissubject matteris 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-dimensionalgraphics andprograms for algebraiccom- 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

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