Abstract Algebra

683 Pages·2007·4.492 MB·English

by  Pierre Antoine Grillet (auth.)

242
Graduate Texts in Mathematics
Editorial Board
S.Axler
K.A.Ribet
Graduate Texts in Mathematics
1 TAKEUTI/ZARING.Introduction to 34 SPITZER.Principles ofRandom Walk.
Axiomatic Set Theory.2nd ed. 2nd ed.
2 OXTOBY.Measure and Category.2nd ed. 35 ALEXANDER/WERMER.Several Complex
3 SCHAEFER.Topological Vector Spaces. Variables and Banach Algebras.3rd ed.
2nd ed. 36 KELLEY/NAMIOKAet al.Linear
4 HILTON/STAMMBACH.A Course in Topological Spaces.
Homological Algebra.2nd ed. 37 MONK.Mathematical Logic.
5 MACLANE.Categories for the Working 38 GRAUERT/FRITZSCHE.Several Complex
Mathematician.2nd ed. Variables.
6 HUGHES/PIPER.Projective Planes. 39 ARVESON.An Invitation to C*-Algebras.
7 J.-P.SERRE.A Course in Arithmetic. 40 KEMENY/SNELL/KNAPP.Denumerable
8 TAKEUTI/ZARING.Axiomatic Set Theory. Markov Chains.2nd ed.
9 HUMPHREYS.Introduction to Lie 41 APOSTOL.Modular Functions and
Algebras and Representation Theory. Dirichlet Series in Number Theory.
10 COHEN.A Course in Simple Homotopy 2nd ed.
Theory. 42 J.-P.SERRE.Linear Representations of
11 CONWAY.Functions ofOne Complex Finite Groups.
Variable I.2nd ed. 43 GILLMAN/JERISON.Rings of
12 BEALS.Advanced Mathematical Analysis. Continuous Functions.
13 ANDERSON/FULLER.Rings and 44 KENDIG.Elementary Algebraic
Categories ofModules.2nd ed. Geometry.
14 GOLUBITSKY/GUILLEMIN.Stable 45 LOÈVE.Probability Theory I.4th ed.
Mappings and Their Singularities. 46 LOÈVE.Probability Theory II.4th ed.
15 BERBERIAN.Lectures in Functional 47 MOISE.Geometric Topology in
Analysis and Operator Theory. Dimensions 2 and 3.
16 WINTER.The Structure ofFields. 48 SACHS/WU.General Relativity for
17 ROSENBLATT.Random Processes.2nd ed. Mathematicians.
18 HALMOS.Measure Theory. 49 GRUENBERG/WEIR.Linear Geometry.
19 HALMOS.A Hilbert Space Problem 2nd ed.
Book.2nd ed. 50 EDWARDS.Fermat's Last Theorem.
20 HUSEMOLLER.Fibre Bundles.3rd ed. 51 KLINGENBERG.A Course in Differential
21 HUMPHREYS.Linear Algebraic Groups. Geometry.
22 BARNES/MACK.An Algebraic 52 HARTSHORNE.Algebraic Geometry.
Introduction to Mathematical Logic. 53 MANIN.A Course in Mathematical Logic.
23 GREUB.Linear Algebra.4th ed. 54 GRAVER/WATKINS.Combinatorics with
24 HOLMES.Geometric Functional Emphasis on the Theory ofGraphs.
Analysis and Its Applications. 55 BROWN/PEARCY.Introduction to
25 HEWITT/STROMBERG.Real and Abstract Operator Theory I:Elements of
Analysis. Functional Analysis.
26 MANES.Algebraic Theories. 56 MASSEY.Algebraic Topology:An
27 KELLEY.General Topology. Introduction.
28 ZARISKI/SAMUEL.Commutative 57 CROWELL/FOX.Introduction to Knot
Algebra.Vol.I. Theory.
29 ZARISKI/SAMUEL.Commutative 58 KOBLITZ.p-adic Numbers,p-adic
Algebra.Vol.II. Analysis,and Zeta-Functions.2nd ed.
30 JACOBSON.Lectures in Abstract Algebra 59 LANG.Cyclotomic Fields.
I.Basic Concepts. 60 ARNOLD.Mathematical Methods in
31 JACOBSON.Lectures in Abstract Algebra Classical Mechanics.2nd ed.
II.Linear Algebra. 61 WHITEHEAD.Elements ofHomotopy
32 JACOBSON.Lectures in Abstract Algebra Theory.
III.Theory ofFields and Galois 62 KARGAPOLOV/MERIZJAKOV.
Theory. Fundamentals ofthe Theory ofGroups.
33 HIRSCH.Differential Topology. 63 BOLLOBAS.Graph Theory.
(continued after index)
Pierre Antoine Grillet
Abstract Algebra
Second Edition
Pierre Antoine Grillet
Dept.Mathematics
Tulane University
New Orleans, LA 70118
USA
[email protected]
Editorial Board
S.Axler K.A.Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco,CA 94132 Berkeley,CA 94720-3840
USA USA
[email protected] [email protected]
Mathematics Subject Classification (2000):20-01 16-01
Library ofCongress Control Number: 2007928732
ISBN-13:978-0-387-71567-4 eISBN-13:978-0-387-71568-1
Printed on acid-free paper.
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Preface
This book is a basic algebra text for first-year graduate students, with some
additionsforthosewhosurviveintoasecondyear. Itassumesthatreadersknow
some linear algebra, and can do simple proofs with sets, elements, mappings,
andequivalencerelations. Otherwise, thematerialisself-contained. Aprevious
semesterofabstractalgebrais,however,highlyrecommended.
Algebratodayisadiverseandexpandingfieldofwhichthestandardcontents
of a first-year course no longer give a faithful picture. Perhaps no single book
can;butenoughadditionaltopicsareincludedheretogivestudentsafaireridea.
Instructors will have some flexibility in devising syllabi or additional courses;
studentsmayreadorpeekattopicsnotcoveredinclass.
Diagrams and universal properties appear early to assist the transition from
proofswithelementstoproofswitharrows;butcategoriesanduniversalalgebras,
which provide conceptual understanding of algebra in general, but require more
maturity, have been placed last. The appendix has rather more set theory than
usual;thisputsZorn’slemmaandcardinalitiesonareasonablyfirmfooting.
Theauthorisfondofsaying(somesay,overlyfond)thatalgebraislikeFrench
pastry: wonderful, but cannot be learned without putting one’s hands to the
dough. Over 1400 exercises will encourage readers to do just that. A few are
simpleproofsfromthetext,placedthereinthebeliefthatusefulfactsmakegood
exercises. Starredproblemsaremoredifficultorhavemoreextensivesolutions.
Algebra owes its name, and its existence as a separate branch of mathemat-
ics, to a ninth-century treatise on quadratic equations, Al-jabr wa’l muqabala,
“the balancing of related quantities”, written by the Persian mathematician al-
Khowarizmi. (TheauthorisindebtedtoProfessorBoumedienneBelkhouchefor
thistranslation.) Algebraretaineditsemphasisonpolynomialequationsuntilwell
into the nineteenth century, then began to diversify. Around 1900, it headed the
revolutionthatmademathematicsabstractandaxiomatic. WilliamBurnsideand
the great German algebraists of the 1920s, most notably Emil Artin, Wolfgang
Krull, andEmmyNoether, usedtheclarityandgeneralityofthenewmathemat-
ics to reach unprecedented depth and to assemble what was then called modern
algebra. The next generation, Garrett Birkhoff, Saunders MacLane, and others,
expanded its scope and depth but did not change its character. This history is
viii Preface
documentedbybriefnotesandreferencestotheoriginalpapers. Timepressures,
sundryevents,andthestateofthelocallibrarieshavekeptthesereferencesabit
short of optimal completeness, but they should suffice to place results in their
historicalcontext,andmayencouragesomereaderstoreadtheoldmasters.
ThisbookisasecondeditionofAlgebra,publishedbythegoodfolksatWiley
in1999. Imeanttoaddafewtopicsandincorporateanumberofusefulcomments,
particularlyfromProfessorGaribaldi,ofEmoryUniversity. Iendeduprewriting
thewholebookfromendtoend.Iamverygratefulforthischancetopolishamajor
work,madepossiblebySpringer,bythepatienceandunderstandingofmyeditor,
Mark Spencer, by the inspired thoroughness of my copy editor, David Kramer,
andbythehospitalityofthepeopleofMarshallandScottsville.
Readerswhoarefamiliarwiththefirstversionwillfindmanydifferences,some
ofthemmajor. Thefirstchaptershavebeenstreamlinedforrapidaccesstosolv-
ability of equations by radicals. Some topics are gone: groups with operators,
Lu¨roth’s theorem, Sturm’s theorem on ordered fields. More have been added:
separabilityoftranscendentalextensions,Hensel’slemma,Gro¨bnerbases,primi-
tiverings,hereditaryrings,ExtandTorandsomeoftheirapplications,subdirect
products. Therearesome450moreexercises. Iapologizeinadvanceforthenew
errors introduced by this process, and hope that readers will be kind enough to
pointthemout.
NewOrleans,Louisiana,andMarshall,Texas,2006.
Contents
Preface............................................................... vii
Starredsectionsandchaptersmaybeskippedatfirstreading.
I.Groups..............................................................1
1. Semigroups ......................................................... 1
2. Groups..............................................................8
3. Subgroups ......................................................... 12
4. Homomorphisms ...................................................18
5. TheIsomorphismTheorems .........................................23
6. FreeGroups........................................................27
7. Presentations.......................................................31
*8. FreeProducts.......................................................37
II.StructureofGroups..................................................43
1. DirectProducts.....................................................43
*2. TheKrull-SchmidtTheorem.........................................48
3. GroupActions......................................................54
4. SymmetricGroups..................................................58
5. TheSylowTheorems................................................64
6. SmallGroups ...................................................... 67
7. CompositionSeries.................................................70
*8. TheGeneralLinearGroup...........................................76
9. SolvableGroups....................................................83
*10. NilpotentGroups...................................................89
*11. SemidirectProducts.................................................92
*12. GroupExtensions...................................................95
III.Rings.............................................................105
1. Rings. ...........................................................105
2. SubringsandIdeals................................................109
3. Homomorphisms. .................................................112
4. DomainsandFields................................................116
5. PolynomialsinOneVariable........................................119
6. PolynomialsinSeveralVariables....................................125
*7. FormalPowerSeries...............................................130
8. PrincipalIdealDomains............................................133
*9. RationalFractions.................................................139
x Contents
10. UniqueFactorizationDomains .....................................141
11. NoetherianRings .................................................146
*12. Gro¨bnerBases ....................................................148
IV.FieldExtensions ..................................................155
1. Fields ............................................................155
2. Extensions........................................................159
3. AlgebraicExtensions ..............................................164
4. TheAlgebraicClosure ............................................ 165
5. SeparableExtensions ..............................................169
6. PurelyInseparableExtensions ......................................173
*7. ResultantsandDiscriminants .......................................176
8. TranscendentalExtensions .........................................181
*9. Separability ......................................................184
V.GaloisTheory ....................................................191
1. SplittingFields ...................................................191
2. NormalExtensions ................................................193
3. GaloisExtensions .................................................197
4. InfiniteGaloisExtensions ..........................................200
5. Polynomials ......................................................204
6. Cyclotomy .......................................................211
7. NormandTrace ..................................................215
8. SolvabilitybyRadicals ............................................221
9. GeometricConstructions ..........................................226
VI.FieldswithOrdersorValuations ....................................231
1. OrderedFields ....................................................231
2. RealFields .......................................................234
3. AbsoluteValues ..................................................239
4. Completions ......................................................243
5. Extensions........................................................247
6. Valuations.........................................................251
7. ExtendingValuations ..............................................256
8. Hensel’sLemma ..................................................261
9. FiltrationsandCompletions ........................................266
VII.CommutativeRings ..............................................273
1. PrimaryDecomposition ...........................................273
2. RingExtensions ..................................................277
3. IntegralExtensions ............................................... 280
4. Localization ......................................................285
5. DedekindDomains ...............................................290
6. AlgebraicIntegers ................................................294
7. GaloisGroups ....................................................297
8. MinimalPrimeIdeals .............................................300
9. KrullDimension ..................................................304
10. AlgebraicSets ....................................................307
Contents xi
11. RegularMappings ................................................310
VIII.Modules ........................................................315
1. Definition .........................................................315
2. Homomorphisms ..................................................320
3. DirectSumsandProducts...........................................324
4. FreeModules ......................................................329
5. VectorSpaces .....................................................334
6. ModulesoverPrincipalIdealDomains ...............................336
7. JordanFormofMatrices ............................................342
8. ChainConditions ..................................................346
*9. Gro¨bnerBases .....................................................350
IX.SemisimpleRingsandModules ....................................359
1. SimpleRingsandModules ......................................... 359
2. SemisimpleModules ...............................................362
3. TheArtin-WedderburnTheorem .....................................366
*4. PrimitiveRings ....................................................370
5. TheJacobsonRadical ..............................................374
6. ArtinianRings .....................................................377
*7. RepresentationsofGroups ..........................................380
*8. Characters ........................................................ 386
*9. ComplexCharacters ................................................389
X.ProjectivesandInjectives ...........................................393
1. ExactSequences ...................................................393
2. PullbacksandPushouts .............................................397
3. ProjectiveModules .................................................401
4. InjectiveModules ..................................................403
*5. TheInjectiveHull ..................................................408
*6. HereditaryRings ...................................................411
XI.Constructions ....................................................415
1. GroupsofHomomorphisms .........................................415
2. PropertiesofHom .................................................419
*3. DirectLimits ......................................................423
*4. InverseLimits .....................................................429
5. TensorProducts....................................................434
6. PropertiesofTensorProducts .......................................441
*7. DualModules .....................................................448
*8. FlatModules ......................................................450
*9. Completions.......................................................456
*XII.ExtandTor ........................................................463
1. Complexes ........................................................463
2. Resolutions .......................................................471
3. DerivedFunctors ..................................................478
4. Ext ...............................................................487
5. Tor ...............................................................493