Advanced modern algebra

1056 Pages·2002·5.644 MB·English

by  Joseph J. Rotman

Advanced Modern Algebra
by Joseph J. Rotman
Hardcover: 1040 pages
Publisher: Prentice Hall; 1st edition (2002); 2nd printing (2003)
Language: English
ISBN: 0130878685
Book Description
This book's organizing principle is the interplay between groups and rings,
where “rings” includes the ideas of modules. It contains basic definitions,
complete and clear theorems (the first with brief sketches of proofs), and
gives attention to the topics of algebraic geometry, computers, homology,
and representations. More than merely a succession of definition-theorem-proofs,
this text put results and ideas in context so that students can appreciate why
a certain topic is being studied, and where definitions originate. Chapter
topics include groups; commutative rings; modules; principal ideal domains;
algebras; cohomology and representations; and homological algebra. For
individuals interested in a self-study guide to learning advanced algebra and
its related topics.
Book Info
Contains basic definitions, complete and clear theorems, and gives attention
to the topics of algebraic geometry, computers, homology, and representations.
For individuals interested in a self-study guide to learning advanced algebra
and its related topics.
Tomywife
Marganit
andourtwowonderfulkids,
DannyandElla,
whomIloveverymuch

Contents
SecondPrinting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
SpecialNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter1 ThingsPast . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. SomeNumberTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. RootsofUnity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3. SomeSetTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter2 GroupsI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2. Permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3. Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4. Lagrange’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5. Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.6. QuotientGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7. GroupActions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Chapter3 CommutativeRingsI . . . . . . . . . . . . . . . . . . . . . 116
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.2. FirstProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.3. Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.4. GreatestCommonDivisors . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.5. Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.6. EuclideanRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.7. LinearAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
VectorSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
LinearTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.8. QuotientRingsandFiniteFields . . . . . . . . . . . . . . . . . . . . . . . 182
v
vi Contents
Chapter4 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
4.1. InsolvabilityoftheQuintic . . . . . . . . . . . . . . . . . . . . . . . . . . 198
FormulasandSolvabilitybyRadicals . . . . . . . . . . . . . . . . . . . 206
TranslationintoGroupTheory . . . . . . . . . . . . . . . . . . . . . . . 210
4.2. FundamentalTheoremofGaloisTheory . . . . . . . . . . . . . . . . . . . 218
Chapter5 GroupsII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.1. FiniteAbelianGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
DirectSums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
BasisTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
FundamentalTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
5.2. TheSylowTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5.3. TheJordan–Ho¨lderTheorem . . . . . . . . . . . . . . . . . . . . . . . . . 278
5.4. ProjectiveUnimodularGroups . . . . . . . . . . . . . . . . . . . . . . . . 289
5.5. Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
5.6. TheNielsen–SchreierTheorem . . . . . . . . . . . . . . . . . . . . . . . . 311
Chapter6 CommutativeRingsII . . . . . . . . . . . . . . . . . . . . . 319
6.1. PrimeIdealsandMaximalIdeals . . . . . . . . . . . . . . . . . . . . . . . 319
6.2. UniqueFactorizationDomains . . . . . . . . . . . . . . . . . . . . . . . . 326
6.3. NoetherianRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
6.4. ApplicationsofZorn’sLemma . . . . . . . . . . . . . . . . . . . . . . . . 345
6.5. Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
6.6. Gro¨bnerBases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
GeneralizedDivisionAlgorithm . . . . . . . . . . . . . . . . . . . . . . 400
Buchberger’sAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Chapter7 ModulesandCategories . . . . . . . . . . . . . . . . . . . 423
7.1. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
7.2. Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
7.3. Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
7.4. FreeModules,Projectives,andInjectives . . . . . . . . . . . . . . . . . . . 471
7.5. GrothendieckGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
7.6. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
Chapter8 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
8.1. NoncommutativeRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
8.2. ChainConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
8.3. SemisimpleRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
8.4. TensorProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
8.5. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
8.6. TheoremsofBurnsideandofFrobenius . . . . . . . . . . . . . . . . . . . 634
Contents vii
Chapter9 AdvancedLinearAlgebra . . . . . . . . . . . . . . . . . . 646
9.1. ModulesoverPIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
9.2. RationalCanonicalForms . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
9.3. JordanCanonicalForms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
9.4. SmithNormalForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
9.5. BilinearForms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
9.6. GradedAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
9.7. DivisionAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727
9.8. ExteriorAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741
9.9. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
9.10. LieAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
Chapter10 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
10.2. SemidirectProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
10.3. GeneralExtensionsandCohomology . . . . . . . . . . . . . . . . . . . . 794
10.4. HomologyFunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813
10.5. DerivedFunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830
10.6. ExtandTor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852
10.7. CohomologyofGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 870
10.8. CrossedProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887
10.9. IntroductiontoSpectralSequences . . . . . . . . . . . . . . . . . . . . . 893
Chapter11 CommutativeRingsIII . . . . . . . . . . . . . . . . . . . 898
11.1. LocalandGlobal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898
11.2. DedekindRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922
Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923
NullstellensatzRedux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
AlgebraicIntegers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938
CharacterizationsofDedekindRings . . . . . . . . . . . . . . . . . . . . 948
FinitelyGeneratedModulesoverDedekindRings . . . . . . . . . . . . . 959
11.3. GlobalDimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
11.4. RegularLocalRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985
Appendix TheAxiomofChoiceandZorn’sLemma . . . . . . . . A-1
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1
Second Printing
Itismygoodfortunethatseveralreadersofthefirstprintingthisbookapprisedmeof
errataIhadnotnoticed,oftengivingsuggestionsforimprovement. Igivespecialthanksto
NickLoehr,RobinChapman,andDavidLeepfortheirgeneroussuchhelp.
PrenticeHallhasallowedmetocorrecteveryerrorfound;thissecondprintingissurely
betterthanthefirstone.
JosephRotman
May2003
viii
Preface
Algebraisusedbyvirtuallyallmathematicians,betheyanalysts,combinatorists,com-
puter scientists, geometers, logicians, number theorists, or topologists. Nowadays, ev-
eryone agrees that some knowledge of linear algebra, groups, and commutative rings is
necessary, and these topics are introduced in undergraduate courses. We continue their
study.
Thisbookcanbeusedasatextforthefirstyearofgraduatealgebra,butitismuchmore
than that. It can also serve more advanced graduate students wishing to learn topics on
theirown;whilenotreachingthefrontiers,thebookdoesprovideasenseofthesuccesses
andmethodsarisinginanarea. Finally,thisisareferencecontainingmanyofthestandard
theoremsanddefinitionsthatusersofalgebraneedtoknow. Thus,thebookisnotonlyan
appetizer,butaheartymealaswell.
WhenIwasastudent, BirkhoffandMacLane’sASurveyofModernAlgebrawasthe
text for my first algebra course, and van der Waerden’s Modern Algebra was the text for
my second course. Both are excellent books (I have called this book Advanced Modern
Algebrainhomagetothem),buttimeshavechangedsincetheirfirstappearance: Birkhoff
and Mac Lane’s book first appeared in 1941, and van der Waerden’s book first appeared
in 1930. There are today major directions that either did not exist over 60 years ago, or
thatwerenotthenrecognizedtobesoimportant. Thesenewdirectionsinvolvealgebraic
geometry, computers, homology, and representations (A Survey of Modern Algebra has
been rewritten as Mac Lane–Birkhoff, Algebra, Macmillan, New York, 1967, and this
versionintroducescategoricalmethods;categorytheoryemergedfromalgebraictopology,
butwasthenusedbyGrothendiecktorevolutionizealgebraicgeometry).
Letmenowaddressreadersandinstructorswhousethebookasatextforabeginning
graduatecourse.IfIcouldassumethateveryonehadalreadyreadmybook,AFirstCourse
inAbstractAlgebra,thentheprerequisitesforthisbookwouldbeplain. Butthisisnota
realisticassumption;differentundergraduatecoursesintroducingabstractalgebraabound,
asdotextsforthesecourses. Formany,linearalgebraconcentratesonmatricesandvector
spacesovertherealnumbers,withanemphasisoncomputingsolutionsoflinearsystems
ofequations;othercoursesmaytreatvectorspacesoverarbitraryfields,aswellasJordan
and rational canonical forms. Some courses discuss the Sylow theorems; some do not;
somecoursesclassifyfinitefields;somedonot.
Toaccommodatereadershavingdifferentbackgrounds,thefirstthreechapterscontain
ix
x Preface
many familiar results, with many proofs merely sketched. The first chapter contains the
fundamental theorem of arithmetic, congruences, De Moivre’s theorem, roots of unity,
cyclotomic polynomials, and some standard notions of set theory, such as equivalence
relationsandverificationofthegroupaxiomsforsymmetricgroups. Thenexttwochap-
ters contain both familiar and unfamiliar material. “New” results, that is, results rarely
taught in a first course, have complete proofs, while proofs of “old” results are usually
sketched. Inmoredetail,Chapter2isanintroductiontogrouptheory,reviewingpermuta-
tions,Lagrange’stheorem,quotientgroups,theisomorphismtheorems,andgroupsacting
on sets. Chapter 3 is an introduction to commutative rings, reviewing domains, fraction
fields,polynomialringsinonevariable,quotientrings,isomorphismtheorems,irreducible
polynomials, finitefields, andsomelinearalgebraoverarbitraryfields. Readersmayuse
“older” portions of these chapters to refresh their memory of this material (and also to
seemynotationalchoices);ontheotherhand,thesechapterscanalsoserveasaguidefor
learningwhatmayhavebeenomittedfromanearliercourse(completeproofscanbefound
inAFirstCourseinAbstractAlgebra). Thisformatgivesmorefreedomtoaninstructor,
for there is a variety of choices for the starting point of a course of lectures, depending
on what best fits the backgrounds of the students in a class. I expect that most instruc-
tors would begin a course somewhere in the middle of Chapter 2 and, afterwards, would
continue from some point in the middle of Chapter 3. Finally, this format is convenient
fortheauthor, becauseitallowsmetoreferbacktotheseearlierresultsinthemidstofa
discussionoraproof. Proofsinsubsequentchaptersarecompleteandarenotsketched.
Ihavetriedtowriteclearandcompleteproofs,omittingonlythosepartsthataretruly
routine; thus, it is not necessary for an instructor to expound every detail in lectures, for
studentsshouldbeabletoreadthetext.
Hereisamoredetailedaccountofthelaterchaptersofthisbook.
Chapter 4 discusses fields, beginning with an introduction to Galois theory, the inter-
relationshipbetweenringsandgroups. Weprovetheinsolvabilityofthegeneralpolyno-
mial of degree 5, the fundamental theorem of Galois theory, and applications, such as a
proofofthefundamentaltheoremofalgebra,andGalois’stheoremthatapolynomialover
afieldofcharacteristic0issolvablebyradicalsifandonlyifitsGaloisgroupisasolvable
group.
Chapter 5 covers finite abelian groups (basis theorem and fundamental theorem), the
Sylowtheorems,Jordan–Ho¨ldertheorem,solvablegroups,simplicityofthelineargroups
PSL(2,k),freegroups,presentations,andtheNielsen–Schreiertheorem(subgroupsoffree
groupsarefree).
Chapter6introducesprimeandmaximalidealsincommutativerings;Gauss’stheorem
thatR[x]isaUFDwhenRisaUFD;Hilbert’sbasistheorem,applicationsofZorn’slemma
to commutative algebra (a proof of the equivalence of Zorn’s lemma and the axiom of
choiceisintheappendix),inseparability,transcendencebases,Lu¨roth’stheorem,affineva-
rieties,includingaproofoftheNullstellensatzforuncountablealgebraicallyclosedfields
(the full Nullstellensatz, for varieties over arbitrary algebraically closed fields, is proved
in Chapter 11); primary decomposition; Gro¨bner bases. Chapters 5 and 6 overlap two
chaptersofAFirstCourseinAbstractAlgebra,butthesechaptersarenotcoveredinmost