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Abstract Algebra PDF

462 Pages·2017·2.07 MB·English
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Abstract Algebra Second Edition Abstract Algebra by Justin R. Smith This is dedicated to my wonderful wife, Brigitte. c2017. JustinR.Smith. Allrightsreserved. (cid:13) ISBN-13: 978-1973766537(Createspaceassigned) ISBN-10: 1973766531 AlsopublishedbyFiveDimensionsPress (cid:3) IntroductiontoAlgebraicGeometry(paperback),JustinSmith. (cid:3) EyeofaFly(Kindleeditionandpaperback),JustinSmith. (cid:3) TheGodVirus,(Kindleeditionandpaperback)byIndigoVoyager. FiveDimensionsPresspage: HTTP://www.five-dimensions.org Email:[email protected] Foreword Algebraistheoffermadebythedeviltothemathematician.Thedevil says“Iwillgiveyouthispowerfulmachine, anditwillanswerany questionyoulike. Allyouneedtodoisgivemeyoursoul; giveup geometryandyouwillhavethismarvelousmachine.” M.F.Atiyah(2001) This book arose out of courses in Abstract Algebra, Galois Theory, Alge- braicGeometry,andManifoldTheorytheauthortaughtatDrexelUniversity. Itisusefulforself-studyorasacoursetextbook. Thefirstfourchaptersaresuitableforthefirstofatwosemesterundergrad- uatecourseinAbstractAlgebraandchaptersfivethroughsevenaresuitablefor thesecondsemester. Chapter 2 on page 5 covers a few preliminaries on basic set theory (ax- iomaticsettheoryisdiscussedinappendixAonpage391). Chapter 3 on page 9 discusses basic number theory, which is a useful in- troduction to the theory of groups. It also presents an application of number theorytocryptography. Chapter4onpage23discussesthetheoryofgroupsandhomomorphisms ofgroups. Chapter 5 on page 93 covers ring-theory with material on computations inpolynomialringsandGröbnerbases. Italsodiscussessomeapplicationsof algebratomotion-planningandrobotics. Chapter 6 on page 139 covers basic linear algebra, determinants, and dis- cussesmodulesasgeneralizationsofvector-spaces. Wealsodiscussresultants ofpolynomialsandeigenvalues. Chapter7onpage209discussesbasicfield-theory,includingalgebraicand transcendental field-extensions, finite fields, and the unsolvability of some classic problems in geometry. This, section 4.9 on page 71, and chapter 9 on page259mightbesuitableforatopicscourseinGaloistheory. Chapter8onpage237discussessomemoreadvancedareasofthetheory ofrings,likeArtinianringsandintegralextensions. Chapter 9 on page 259 covers basic Galois Theory and should be read in conjunctionwithchapter7onpage209andsection4.9onpage71. Chapter10onpage287discussesdivision-algebrasovertherealnumbers andtheirapplications. Inparticular,itdiscussesapplicationsofquaternionsto computer graphics and proves the Frobenius Theorem classifying associative divisionalgebrasoverthereals. Italsodevelopsoctonionsanddiscussestheir properties. vii Chapter11onpage303givesanintroductiontoCategoryTheoryandap- pliesittoconceptslikedirectandinverselimitsandmultilinearalgebra(tensor productsanexterioralgebras). Chapter 12 on page 351 gives a brief introduction to algebraic geometry andprovesHilbert’sNullstellensatz. Chapter13onpage363discussessome20th centurymathematics: homol- ogyandcohomology.Itcoverschaincomplexesandchain-homotopyclassesof mapsandculminatesinalittlegroup-cohomology,includingtheclassification ofabelianextensionsofgroups. (cid:127) Sections marked in this manner are more advanced or specialized and may be skippedonafirstreading. (cid:127)(cid:127) Sections marked in this manner are even more advanced or specialized and maybeskippedonafirstreading(orskippedentirely). IamgratefultoMatthiasEttrichandthemanyotherdevelopersofthesoft- ware,LYX—afreefrontendtoLATEXthathastheeaseofuseofawordproces- sor, with spell-checking, an excellent equation editor, and a thesaurus. I have usedthissoftwareforyearsandthecurrentversionismorepolishedandbug- freethanmostcommercialsoftware. LYXisavailablefromHTTP://www.lyx.org. Contents Foreword vii ListofFigures xiii Chapter1. Introduction 1 Chapter2. Preliminaries 5 2.1. Settheory 5 2.2. Operationsonsets 5 2.3. ThePowerSet 6 Chapter3. Aglimpseofnumbertheory 9 3.1. Primenumbersanduniquefactorization 9 3.2. Modulararithmetic 14 3.3. TheEulerφ-function 17 3.4. Applicationstocryptography 21 Chapter4. GroupTheory 23 4.1. Introduction 23 4.2. Homomorphisms 29 4.3. Cyclicgroups 30 4.4. Subgroupsandcosets 33 4.5. SymmetricGroups 38 4.6. AbelianGroups 47 4.7. Group-actions 61 4.8. TheSylowTheorems 68 4.9. Subnormalseries 71 4.10. FreeGroups 74 4.11. Groupsofsmallorder 90 Chapter5. TheTheoryofRings 93 5.1. Basicconcepts 93 5.2. Homomorphismsandideals 96 5.3. IntegraldomainsandEuclideanRings 102 5.4. Noetherianrings 106 5.5. Polynomialrings 110 5.6. Uniquefactorizationdomains 129 Chapter6. ModulesandVectorSpaces 139 6.1. Introduction 139 6.2. Vectorspaces 141 ix x CONTENTS 6.3. Modules 191 6.4. Ringsandmodulesoffractions 204 Chapter7. Fields 209 7.1. Definitions 209 7.2. Algebraicextensionsoffields 214 7.3. ComputingMinimalpolynomials 219 7.4. Algebraicallyclosedfields 224 7.5. Finitefields 228 7.6. Transcendentalextensions 231 Chapter8. Furthertopicsinringtheory 237 8.1. Artinianrings 237 8.2. Integralextensionsofrings 240 8.3. TheJacobsonradicalandJacobsonrings 248 8.4. Discretevaluationrings 251 8.5. Gradedringsandmodules 255 Chapter9. GaloisTheory 259 9.1. BeforeGalois 259 9.2. Galois 261 9.3. Isomorphismsoffields 262 9.4. RootsofUnity 265 9.5. CyclotomicPolynomials 266 9.6. Groupcharacters 269 9.7. GaloisExtensions 272 9.8. Solvabilitybyradicals 278 9.9. Galois’sGreatTheorem 281 9.10. Thefundamentaltheoremofalgebra 285 Chapter10. DivisionAlgebrasoverR 287 10.1. TheCayley-DicksonConstruction 287 10.2. Quaternions 291 10.3. Octonionsandbeyond 297 Chapter11. Atasteofcategorytheory 303 11.1. Introduction 303 11.2. Functors 309 11.3. Adjointfunctors 313 11.4. Limits 315 11.5. Abeliancategories 324 11.6. Tensorproducts 328 11.7. TensorAlgebrasandvariants 341 Chapter12. Alittlealgebraicgeometry 351 12.1. Introduction 351 12.2. Hilbert’sNullstellensatz 355 Chapter13. Cohomology 363 13.1. Chaincomplexesandcohomology 363

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