Abstract Algebra

462 Pages·2017·2.07 MB·English

by  Justin R. Smith

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