Algebra. Fields with structure, algebras and advanced topics PDF
Preview Algebra. Fields with structure, algebras and advanced topics
Universitext Editorial Board (North America):: S. Axler K.A. Ribet Universitext Aguilar, M.; Gitler, S.; Prieto, C.: Algebraic Blowey, J.F.; Craig, A.; Shardlow, T. (Eds.): TopologyfromaHomotopicalViewpoint Frontiers in Numerical Analysis, Durham Aksoy,A.;Khamsi,M.A.: MethodsinFixed 2002,andDurham2004 PointTheory Blyth,T.S.: LatticesandOrderedAlgebraic Structures Alevras, D.; Padberg M.W.: Linear Opti- mizationandExtensions Bo¨rger, E.; Gra¨del, E.; Gurevich, Y.: The ClassicalDecisionProblem Andersson,M.: TopicsinComplexAnalysis Bo¨ttcher, A; Silbermann, B.: Introduction Aoki, M.: State Space Modeling of Time toLargeTruncatedToeplitzMatrices Series Boltyanski, V.; Martini, H.; Soltan, P.S.: Arnold,V.I.:LecturesonPartialDifferential ExcursionsintoCombinatorialGeometry Equations Boltyanskii, V.G.; Efremovich, V.A.: Intu- Arnold,V.I.;Cooke,R.: OrdinaryDifferen- itiveCombinatorialTopology tialEquations Bonnans, J.F.; Gilbert, J.C.; Lemarchal, C.; Audin,M.: Geometry Sagastizbal,C.A.: NumericalOptimization Aupetit,B.: APrimeronSpectralTheory Booss, B.; Bleecker, D.D.: Topology and Bachem,A.;Kern,W.: LinearProgramming Analysis Duality Borkar,V.S.: ProbabilityTheory Bachmann, G.; Narici, L.; Beckenstein, E.: BruntB.van: TheCalculusofVariations FourierandWaveletAnalysis Bu¨hlmann,H.;Gisler,A.: ACourseinCred- Badescu,L.: AlgebraicSurfaces ibilityTheoryanditsApplications Balakrishnan,R.;Ranganathan,K.: AText- Carleson, L.; Gamelin, T.W.: Complex bookofGraphTheory Dynamics Balser, W.: Formal Power Series and Cecil, T.E.: Lie Sphere Geometry: With Linear Systems of Meromorphic Ordinary ApplicationsofSubmanifolds DifferentialEquations Chae,S.B.: LebesgueIntegration Bapat, R.B.: Linear Algebra and Linear Chandrasekharan, K.: Classical Fourier Models Transform Benedetti, R.; Petronio, C.: Lectures on Charlap, L.S.: BieberbachGroupsandFlat HyperbolicGeometry Manifolds Benth,F.E.: OptionTheorywithStochastic Chern, S.: Complex Manifolds without Analysis PotentialTheory Berberian, S.K.: Fundamentals of Real Chorin, A.J.; Marsden, J.E.: Mathematical Analysis IntroductiontoFluidMechanics Berger,M.: GeometryI,andII Cohn,H.:AClassicalInvitationtoAlgebraic Bhattacharya, R.; Waymire, E.C.: A Basic NumbersandClassFields CourseinProbabilityTheory Curtis,M.L.: AbstractLinearAlgebra Bliedtner,J.;Hansen,W.: PotentialTheory Curtis,M.L.: MatrixGroups Blowey, J.F.; Coleman, J.P.; Craig, A.W. Cyganowski, S.; Kloeden, P.; Ombach, J.: (Eds.): Theory and Numerics of Differen- From Elementary Probability to Stochastic tialEquations DifferentialEquationswithMAPLE (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 [email protected] 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 [email protected] [email protected] 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 © 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com “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.
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