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[美] 爱德华滋 著 / 世界图书出版公司 / 2010-09 / 平装
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上书时间2020-11-26
数学研究生教材:伽罗瓦理论
ThisexpositionofGaloistheorywasoriginallygoingtobeChapterIofthecontinuationofmybookFerrnatsLastTheorem,butitsoonoutgrewanyreasonableboundsforanintroductorychapter,andIdecidedtomakeitaseparatebook.However,thisdecisionwaspromptedbymorethanjustthelength.Followingthepreceptsofmysermon"ReadtheMasters!"[E2],ImadethereadingofGaloisoriginalmemoiramajorpartofmystudyofGaloistheory,andIsawthatthemoderntreatmentsofGaloistheorylackedmuchofthesimplicityandclarityoftheoriginal.ThereforeIwantedtowriteaboutthetheoryinawaythatwouldnotonlyexplainit,butexplainitintermscloseenoughtoGaloisowntomakehismemoiraccessibletothereader,inthesamewaythatItriedtomakeRiemannsmemoironthezetafunctionandKummerspapersonFermatsLastTheoremaccessibleinmyearlierbooks,[Eliand[E3].ClearlyIcouldnotdothiswithintheconfinesofoneexpositorychapter
Acknowledgments1.Galois2.InfluenceofLagrange3.Quadraticequations4.1700B.c.toA.D.15005.Solutionofcubic6.Solutionofquartic7.Impossibilityofquintic8.Newton9.Symmetricpolynomialsinroots10.Fundamentaltheoremonsymmetricpolynomials11.Proof12.Newtonstheorem13.DiscriminantsFirstExerciseSet1314.Solutionofcubic15.LagrangeandVandermonde16.Lagrangeresolvents17.Solutionofquarticagain18.Attemptatquintic19.LagrangesRdflexionsSecondExerciseSet2220.Cyclotomicequations21.Thecasesn=3,522.n=7,1123.Generalcase24.Twolemmas25.Gausssmethod26.p-gonsbyrulerandcompass27.SummaryThirdExerciseSet3128.Resolvents29.Lagrangestheorem30.Proof31.Galoisresolvents32.ExistenceofGaloisresolvents33.RepresentationofthesplittingfieldasK(t)34.Simplealgebraicextensions35.Euclideanalgorithm36.Constructionofsimplealgebraicextensions37.GaloismethodFourthExerciseSet4538.Review39.Finitepermutationgroups40.Subgroups,normalsubgroups41.TheGaloisgroupofanequation42.ExamplesFifthExerciseSet563.Solvabilitybyradicals44.ReductionoftheGaloisgroupbyacyclicextension5.Solvablegroups46.Reductiontoanormalsubgroupofindexp47.Theoremonsolutionbyradicals(assumingrootsofunity)48.SummarySixthExerciseSet6549.Splittingfields50.Fundamentaltheoremofalgebra(so-called)51.Constructionofasplittingfield52.Needforafactorizationmethod53.Threetheoremsonfactorizationmethods54.Uniquenessoffactorizationofpolynomials55.Factorizationover56.Over57.Gaussslemma,factorizationover58.Overtranscendentalextensions59.Ofpolynomialsintwovariables60.Overalgebraicextensions61.FinalremarksSeventhExerciseSet8162.ReviewofGaloistheory63.FundamentaltheoremofGaloistheory(so-called)64.Galoisgroupofxp-1=0over65.Solvabilityofthecyclotomicequation6.Theoremonsolutionbyradicals67.Equationswithliteralcoefficients68.Equationsofprimedegree69.Galoisgroupofxn-1=0over70.Proofofthemainproposition71.DeductionofLemma2of24EighthExerciseSet97Appendix1.MemoirontheConditionsforSolvabilityofEquationsbyRadicals,byEvaristeGaloisAppendix2.SynopsisAppendix3.GroupsAnswerstoExercisesListofExercisesReferencesIndex
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