Algebra. Fields with structure, algebras and advanced topics

343 Pages·2007·2.21 MB·English

by  Falko Lorenz

Universitext

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K.A. Ribet
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(continuedaftertheindex)
Falko Lorenz
Algebra
Volume II:
Fields with Structure, Algebras
and Advanced Topics
W ith the collaboration of the translator, Silvio Levy
ABC
Falko Lorenz
FB Mathematik Institute
University Mü nster
Mü nster, 48149
Germany
[e-mail]

Editorial Board
(North America):
S. Axler K.A. Ribet
Mathematics Department Mathem atics Department
San Fran cisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley , CA 94720-3840
USA USA
[e-mail] [e-mail]

ISBN: 978-0-387-72487-4 e-ISBN: 978-0-387-72488-1
Library of Congress Control Num ber: 200593 2557
Mathematics Subject Classification (2000): 11-01,12-01,13-01
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“Becausecertaintyisdesirableindidacticdiscourse
—thepupilwishestohavenothinguncertaindeliv-
ered to him—the teacher cannot let any problem
stand,circlingitfromadistance,sotospeak.Things
mustbedeterminedatonce.stakedout,astheDutch
say1/,andsoonebelievesforawhilethatoneowns
theunknownterritory,untilanotherpersonripsout
the stakes again and immediately sets them down,
nearerorfartherasthecasemaybe,onceagain.”
J.W.Goethe,inWerke(Weimar1893),partII,vol.11
(“ScienceinGeneral”),p.133.
Foreword
InthissecondvolumeofAlgebra,Ihavefollowedthesameexpositoryguidelines
laid out in the preface to the first volume, with the difference that now pedagogic
considerationscantakeasecondaryroleinfavorofamorematureviewpointonthe
content.
I imagine the reader of this second volume to be a student who already has a
working knowledge of algebra and is eager to extend and deepen this knowledge
in one direction or another. Thus, in sections that can to a large extent be studied
independentlyofthe rest,I havemadea broaderchoiceofpresentation.
There was good reason, in my opinion, to let the material in the first volume
be guided by an emphasis on fields. Thus it was natural to present in this second
volume certain classes of fields having additionalstructure. Among these we deal
firstwithorderedfields,in parttoarouseinterestin theareaofrealalgebra,which
isgivenshortshriftinmostcurrenttextbooks(thoughitwasmuchesteemedinthe
nineteenth century and gained new momentum in the 1920s through the work of
ArtinandSchreier). Italsoseemedworthbroachingcertainaspectsofthetheoryof
quadraticforms.
Next, special attention is devoted to the theory of valued fields. Local fields
representtoday,overahundredyearsaftertheirdiscoverybyHensel,acompletely
standardprerequisiteinmanyareasofmathematics.
Besides making an effort not to treat superficially any area once selected for
coverage, I also aimed for some diversity. Thus I decided not to stay within the
confines of field theory proper, but rather to include another major theory, that of
semisimple algebras. In this context it seemed a matter of course to discuss the
rudimentsof finite group representationsas well. This path to the subject, offered
here instead of the more directone openedup by Schur, is well worth the trouble,
1Goethewritesthepast participleof theDutchverbbepalen, bridgingthetwomeanings:
literally‘toplantstakes’,butinitsnormalusage‘toprescribe,determine,fix,setinplace.’
vi Foreword
especiallyifoneisinterestedalsoinquestionsofrationalityinrepresentationtheory,
astheyaretreatedatthe endofthebook.
Undoubtedlythisvolumecontainsmorematerialthancanbecoveredinoneun-
dergraduatesemester. Somesectionsareperhapssuitableforintroductorygraduate
seminars. Several topics absent from average textbooks are included here, as they
fit naturally with our treatment: among them we mention the Witt calculus, Tsen
ranktheory,andlocalclassfield theory.
Mywarmthanksgotoallwhohelpedinthecreationofthisbook: thestudents
in my course,fortheir invigoratinginterest; my facultycolleagues,formuchgood
advice and for prodding me on with their frequent inquiries about when the book
wouldbeready;FlorianPop,foraconversationinHeidelberg,whichpersuadedme
to include the topic of local class field theory; Hans Daldrop, BurkhardtDorn and
HubertSchulzeRelau,fortheircriticalreadingoflargeportionsofthemanuscript,
andthelatteralsoforhiscarefulworkontheindex;BernadetteBourscheidforher
efficientpreparationoftheoriginaltypescript;thepublishersofthefirstGermanedi-
tion(1990),BI-Wissenschaftsverlag,fortheirrenewedcooperation,andparticularly
theeditorHermannEngesser,forunderstandingandpatientadvice.
In the preparationof the second German edition (1997)I again benefited from
suggestions, praise and criticism from colleagues, including S. Böge, B. Huppert,
J.Neukirch,P.RoquetteandK.Wingberg,andfromtheinvolvementofstudents—
notonlyfromandmycoursebutalsofromelsewhere—whosewatchfulreadingled
toimprovements. SpecialthanksgotoSusanneBosseforherprofessionalresetting
ofthetextin LATEX.
Now I am pleasedto see thissecondvolumeofmy workbeingmadeavailable
in English as well. I’m very thankful to Springer New York and its mathematics
editor,MarkSpencer,forhissupportandgoodadvice. Thetranslation,like thatof
thefirstvolume,wasdonebySilvioLevy,andonceagainhehassuggestedhelpful
improvements to the exposition. I shall look back upon this fruitful collaboration
withfondnessandappreciation.
Münster,September2007 FalkoLorenz
Contents
Foreword.......................................................... v
20 OrderedFieldsandRealFields .................................. 1
Orderedandpreorderedfields..................................... 2
Extensionsoffield orders ........................................ 4
Real-closedfields ............................................... 5
Thefundamentaltheoremofalgebra............................... 7
Artin’scharacterizationof real-closedfields......................... 8
Sylvester’stheoremonthenumberofrealroots ..................... 10
Extensionoforder-preservinghomomorphisms...................... 11
Existenceofrealspecializations................................... 12
21 Hilbert’sSeventeenthProblemandtheRealNullstellensatz ......... 15
Artin’ssolutiontoHilbert’sseventeenthproblem .................... 15
GeneralizationtoaffineK-varieties ................................ 16
TherealNullstellensatz .......................................... 18
Positivedefinitefunctionsonsemialgebraicsets ..................... 23
Positivedefinitesymmetricfunctions .............................. 25
22 OrdersandQuadraticForms.................................... 29
Wittequivalence;the WittringW.K/anditsprimeideals ............ 30
ThespectrumofW.K/ .......................................... 32
Thetorsionelementsof W.K/ ................................... 34
ThezerodivisorsofW.K/....................................... 37
23 AbsoluteValuesonFields ....................................... 39
Absolutevaluesonthe rationals................................... 40
Nonarchimedeanabsolutevalues .................................. 45
Completionofabsolutevalues .................................... 47
Thefield(cid:2) ofp-adic numbers................................... 52
p
Equivalenceofnorms............................................ 54
Hensel’sLemma................................................ 56
Extensionofabsolutevalues...................................... 59
viii Contents
24 ResidueClassDegreeandRamificationIndex ..................... 65
Discreteabsolutevalues.......................................... 67
Theformulaef Dnforcomplete,discretevaluations ................ 68
Unramifiedextensions ........................................... 70
Purelyramifiedextensions ....................................... 73
25 LocalFields ................................................... 75
Classification oflocalfields ...................................... 76
Connectionwithglobalfields ..................................... 78
Completionofthealgebraicclosureof(cid:2) .......................... 80
p
SolvabilityofGaloisgroups ...................................... 81
Structureofthe multiplicativegroup ............................... 83
Thecase of(cid:2) ................................................. 91
p
26 WittVectors................................................... 93
Teichmüllerrepresentatives....................................... 94
Ghostcomponents .............................................. 96
Witt’sLemma .................................................. 98
TheringofWittvectors ......................................... 100
HigherArtin–Schreiertheory ..................................... 105
27 TheTsenRankofaField ....................................... 109
Tsen’stheorems ................................................ 110
Behaviorof theTsenrankwithrespectto extensions................. 112
Normforms.................................................... 114
C -fields....................................................... 116
i
TheLang–NagataTheorem....................................... 118
Finite fieldshaveTsenrank1..................................... 121
TheChevalley–WarningTheorem ................................. 122
AlgebraicallyclosedfieldshaveTsen rank0 ........................ 123
Krull’sdimensiontheorem ....................................... 124
28 FundamentalsofModules....................................... 127
Fundamentalsoflinearalgebra.................................... 128
Simpleandsemisimplemodules .................................. 133
Noetherianandartinianmodules .................................. 139
TheJordan–HölderTheorem ..................................... 142
TheKrull–Remak–SchmidtTheorem .............................. 143
TheJacobsonradical ............................................ 144
Nilpotenceoftheradicalin artinianrings........................... 148
Artinianalgebrasarenoetherian................................... 149
29 WedderburnTheory............................................ 151
Simplealgebras................................................. 151
Decompositionofsimple algebras................................. 152
Wedderburn’sstructuretheorem................................... 155
Tensorproductsofsimple algebras ................................ 160
TheBrauergroupofa field....................................... 164
Contents ix
Tensorproductsofsemisimple algebras ............................ 166
TheCentralizerTheoremandsplittingfields ........................ 168
TheSkolem–NoetherTheorem.................................... 173
Reducednormandtrace ......................................... 176
30 CrossedProducts .............................................. 183
TherelativeBrauergroupofGaloisextensions ...................... 184
Inflationandrestriction .......................................... 190
TheBrauergroupis atorsiongroup ............................... 194
Cyclic algebras ................................................. 196
Quaternionalgebras ............................................. 203
Cohomologygroupsandtheconnectinghomomorphism.............. 212
Corestriction ................................................... 217
31 TheBrauerGroupofaLocalField............................... 223
Existenceofunramifiedsplittingfields ............................. 224
TheequalityeDf Dnforlocaldivisionalgebras................... 226
TherelativeBrauergroupinthe unramifiedcase..................... 229
TheHasse invariant ............................................. 231
Consequences .................................................. 234
32 LocalClassFieldTheory........................................ 239
Thelocalnormresiduesymbol ................................... 240
Functorialpropertiesofthenormresiduesymbol .................... 242
Thelocalreciprocitylaw......................................... 243
Thegroupof universalnormsistrivial ............................. 247
Thelocalexistencetheorem ...................................... 249
ThelocalKronecker–Webertheorem............................... 251
33 SemisimpleRepresentationsofFiniteGroups ..................... 253
Terminology ................................................... 254
Maschke’sTheorem............................................. 259
ApplicationsofWedderburntheory ................................ 261
Orthogonalityrelationsforcharacters .............................. 265
Integralitypropertiesofcharacters................................. 268
Inducedrepresentationsandinducedcharacters...................... 272
Artin’sinductiontheorem ........................................ 275
TheBrauer–Wittinductiontheorem................................ 277
34 TheSchurGroupofaField ..................................... 283
Schurindexofabsolutelyirreduciblecharacters ..................... 284
Schuralgebras.................................................. 286
Reductionto cyclotomicalgebras.................................. 288
TheSchurgroupofa localfield................................... 294
Appendix: ProblemsandRemarks ................................... 299
Index ............................................................. 331
20
Ordered Fields and Real Fields
1. As afirstclass offieldswithadditionalstructurewenowturntoorderedfields.
Definition1. LetK beafield. Givenanorderrelation(cid:2)onthesetK,wesaythat
K—or,moreformally,the pair.K;(cid:2)/—isan orderedfield if
(1) (cid:2) isa totalorder.
(2) a(cid:2)b impliesaCc(cid:2)bCc.
(3) a(cid:2)b and0(cid:2)c implyac(cid:2)bc.
In this situation we also say that (cid:2) is a fieldorder—or, if no confusioncan arise,
justanorder—onK.
If (cid:2) is an order on a set, we write a<b if a(cid:2)b and a¤b. Often instead of
a(cid:2)b ora<b wewriteb(cid:3)aorb>a. Becauseof(2)wehaveinanorderedfield
a(cid:2)b (cid:4) b(cid:5)a(cid:3)0:
Thustheorder(cid:2) ofanorderedfieldK isentirelydeterminedbytheset
P Dfa2Kja(cid:3)0g;
calledthepositiveset (orsetofpositiveelements)of(cid:2). We have
(4) PCP (cid:6)P andPP (cid:6)P;
(5) P\(cid:5)P Df0g;
(6) P[(cid:5)P DK.
Conversely, if P is a subset of a field K satisfying properties (4), (5) and (6),
therelation(cid:2)definedby
a(cid:2)b ” b(cid:5)a2P
makesK intoanorderedfield(withpositiveset P). Forthisreasonwesometimes
alsocallsucha subsetP ofK anorder ofK.