Some Topics in Algebra: An Advanced Undergraduate Course at PKU

209 Pages·2014·1.758 MB·English

by  Michel Broué (auth.)

Mathematical Lectures from Peking University
Michel Broué
Some Topics
in Algebra
An Advanced Undergraduate Course
at PKU
Mathematical Lectures from Peking University
Forfurthervolumes:
www.springer.com/series/11574
Michel Broué
Some Topics in Algebra
An Advanced Undergraduate Course at PKU
MichelBroué
InstitutUniversitairedeFrance
UniversitéParisDiderot—Paris7
Paris,France
ISSN2197-4209 ISSN2197-4217(electronic)
ISBN978-3-642-41268-4 ISBN978-3-642-41269-1(eBook)
DOI10.1007/978-3-642-41269-1
SpringerHeidelbergNewYorkDordrechtLondon
MathematicsSubjectClassification: 11-01,12-01,13-01
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Preface
Never, before I had lectured in front of second and third years students of Peking
University, did I feel that strongly how much mathematics are universal, a world
wherehumanmindsthinkalike.AndneverbeforehadIenjoyeditsomuch.
The course I was supposed to give (an advanced undergraduate introduction to
Algebra) had no specific syllabus. I decided to let it go, pushed or pulled by the
student’s reactions and my own feelings. It turned out to become an amazing and
delightfulencounterbetween,ononehand,ideasanddiscoveriesofGermanmath-
ematicians from the end of the XIXth and the beginning of the XXth centuries1
revisitedandtaughtbysomeFrenchmathematiciansfromtheXXthcentury2,and,
ontheotherhand,youngbrilliantChinesestudentsoftheXXIstcentury.
The pleasure of these students while discoveringthese concepts,results, exam-
ples,hasbeenobviousallalongthecourse,andevensometimesexpressedloudly.
Moreover,thespeedoftheirunderstandingandhandlingnotionswhichweremostly
newtothemwasamazing.Acoupleoftimes,attheintermission,oneofthemcame
and politely told me that he thought he had found a more elegant proof than the
oneIhadjustgiven—andeachtimehewasindeedright,hisproofwasbetter,more
elegant,morenatural.
Elegant, efficient, natural, pertinent, beautiful, clever, exciting: these are words
sometimesheardwhenamathematiciandiscoversanewapproach,anewproof,or
even a new version of an old result. Whatever country, origin, culture that mathe-
maticianmaybefrom:whatisbeautifulandpertinentforaGermanHerrProfessor
of the XIXth Century is also beautiful and pertinent for a young Chinese student
of2013.Ofcourse,universalityisnotthepeculiarityofmathematics,itiscertainly
shared by most of the arts, and partly by philosophy. But the essence of the uni-
versality of mathematics is not directly connected with feelings and events of any
humanlife,painorjoy,loveordisaster,war,freedom,deathorfuture.Besides,the
1IdealswerefirstdefinedbyRichardDedekindin1876inthethirdeditionofhisbook“Vorlesun-
genüberZahlentheorie”(LecturesonNumberTheory),afterErnstKummerhadintroducedthe
conceptof“idealnumbers”.ThenotionwaslaterexpandedbyDavidHilbertandEmmyNoether.
2LikeNicolasBourbaki.
v
vi Preface
universalityofmathematicsisarule,almostatheorem:whatisconsideredbyallas
goodisindeedgood.Idothinkthisisoneofthewondersoftheworldwelivein.
Themoreelegantproofsofthestudentsareintegratedinthisbook,withoutquo-
tationtotheirauthorssinceIdidnotknowtheirnames.Thisisoneofthereasons
whythebookisdedicatedtothestudentsofPKU.
CONVERSATIONBETWEENMATHEMATICIANS—©AnoukGrinberg
Preface vii
Abstract
During the Springs of 2011 and 2012, I was invited by the Beijing International
CenterforMathematicsResearchtogiveanadvancedundergraduatealgebracourse
(once a week over two months each year). This is part of the Everest project of
ChineseEducationMinistryonfirstclassstudentstraining.
Thisbookhasbeenwrittenduringandforthatcourse.Bynowaydoesitpretend
toanytypeofexhaustivity.Itisa quickandcontingentintroductiontoAlgebrain
front of an extremely pleasant and passionate audience, heterogeneous but persis-
tent.Itcertainlyreflectssomeofmyowntastes,andmainlytheconstraintsofsuch
ashortperiodofteaching.
Aremarkaboutthelasttwosections:followingawellestablishedtradition,we
hadplannedtoconcludebylecturingonthestructureoffinitelygeneratedmodules
overprincipalidealdomains.Butduringtheprocessofthecourse,afterexplaining
thatthenotionofprojectivemoduleissomehowmorenaturalthanthenotionoffree
module,itbecameratherinevitabletoreplaceprincipalidealdomainsbyDedekind
rings;thisislesstraditionalintheliterature—butnotreallymoredifficult.
viii Preface
Prerequisites
Thisbookrequiresacertainfamiliaritywiththenotionsofgroups,rings,fields,and
speciallywiththeundergraduateknowledgeoflinearalgebra.Morespecifically,let
kbeacommutativefield.Weassumethereaderknows
• thedefinitionoftheringofpolynomialk[X ,...,X ]innindeterminates,
1 n
• the Euclidean division in Z and in k[X], as well as some of the consequences,
like: both these rings are principal ideal domains, hence for p a prime number
andP(X)anirreduciblepolynomial,bothquotientsZ/pZandk[X]/(P(X))are
fields;
• themainresultsofanundergraduatecourseonk-linearalgebra;
• matricesandtheirdeterminants.
Thefollowingidentitywillnotbeproved:letMbeann×nmatrixwithentries
in k, let tCom(M) denote the transpose of its matrix of cofactors, let 1 be the
n
identityn×nmatrix;then
tCom(M).M=det(M).1 .
n
WetakeforgrantedthatthereaderisfamiliarwiththestandardnotationN(for
“numbers”)—note that by convention N={0,1,2,...}, Z for “Zahlen”), Q (for
“quotients”), R (for “reals”), C (for “complexes”), as well as F = Z/pZ (for
p
“finite”)—see[7],p.3.
Byconvention,afield isacommutativeringwhereallnonzeroelementsarein-
vertible.Anoncommutativeringwhereallnonzeroelementsareinvertibleiscalled
adivisionring.
Weshallalsousethefollowingnotation.
• ForΩ anyfiniteset,|Ω|willdenotethenumberofitselements.
• Foranysequenceξ ,...,ξ ,oranyproductξ ···ξ ,andforallj =1,...,n,we
1 n 1 n
set(withobviousadhocconventionforξ0 andξn+1):
(ξ1,...,(cid:2)ξj,...,ξn):=(ξ1,...,ξj−1,ξj+1,...,ξn), and
ξ1···(cid:2)ξj···ξn:=ξ1···ξj−1ξj+1···ξn.
(cid:2)
A subset (subgroup, subring, submodule,...) Ω of a set (group, ring, mod-
ule,...)Ω issaidtobeproperifΩ(cid:2)(cid:3)=Ω.
Acknowledgements
IwarmlythankYanjunLiu,XiongHuan,andspeciallyGunterMalle,fortheircare-
fulandpatientreadingsandcorrectionsofthemanuscript.Mythanksgoalsotothe
students who attended the courses which motivated these notes, for their interest,
theirattention,theirquestions,theirwonderforMathematics.
ix
Contents
1 RingsandPolynomialAlgebras . . . . . . . . . . . . . . . . . . . . 1
1.1 FirstDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 CanonicalMorphisms . . . . . . . . . . . . . . . . . . . 9
1.1.3 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.4 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.5 Fields,DivisionRings . . . . . . . . . . . . . . . . . . . 20
1.2 PrimeandMaximalIdeals,IntegralDomains . . . . . . . . . . . 36
1.2.1 DefinitionandFirstExamples . . . . . . . . . . . . . . . 36
1.2.2 ExamplesinPolynomialRings . . . . . . . . . . . . . . 37
1.2.3 NilradicalandRadical . . . . . . . . . . . . . . . . . . . 39
1.2.4 IntegralDomains,FieldsofFractions . . . . . . . . . . . 42
1.3 DivisibilityandIrreducibleElements . . . . . . . . . . . . . . . 47
1.3.1 DivisorsandIrreducibleElements. . . . . . . . . . . . . 47
1.3.2 EuclideanRings . . . . . . . . . . . . . . . . . . . . . . 49
1.3.3 GCDandLCM . . . . . . . . . . . . . . . . . . . . . . 51
1.3.4 CaseofZ[i]andApplication . . . . . . . . . . . . . . . 52
1.3.5 IrreducibilityCriteriainR[X] . . . . . . . . . . . . . . . 54
1.4 PolynomialRingsinSeveralIndeterminates . . . . . . . . . . . 66
1.4.1 UniversalProperty,Substitutions . . . . . . . . . . . . . 66
1.4.2 TransferProperties. . . . . . . . . . . . . . . . . . . . . 70
1.4.3 SymmetricPolynomials . . . . . . . . . . . . . . . . . . 71
1.4.4 ResultantandDiscriminant . . . . . . . . . . . . . . . . 76
2 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.1 DefinitionsandConventions . . . . . . . . . . . . . . . . . . . . 87
2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.1.2 Submodules . . . . . . . . . . . . . . . . . . . . . . . . 90
2.1.3 TorsionElements,TorsionSubmodule . . . . . . . . . . 96
2.1.4 FreeandGeneratingSystems,FreeModules . . . . . . . 98
2.1.5 Constructions:DirectSums,Products,TensorProducts . 102
xi