fileds and galois theory [jnl article] - j. milne

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1、FIELDSANDGALOISTHEORYJ.S.MILNEAbstract.ThesearethenotesforthesecondpartofMath594,UniversityofMichigan,Winter1994,exactlyastheywerehandedoutduringthecourseexceptforsomeminorcorrections.Pleasesendcommentsandcorrectionstomeatjmilne@umich.eduusing“Math594”asthesubject.v2.01(August21,1996).F

2、irstversionontheweb.v2.02(May27,1998).About40minorcorrections(thankstoHenryKim).Contents1.ExtensionsofFields11.1.Definitions11.2.Thecharacteristicofafield11.3.ThepolynomialringF[X]21.4.Factoringpolynomials21.5.Extensionfields;degrees41.6.Constructionofsomeextensions41.7.Generatorsofextensi

3、onfields51.8.Algebraicandtranscendentalelements61.9.Transcendentalnumbers81.10.Constructionswithstraight-edgeandcompass.92.SplittingFields;AlgebraicClosures122.1.Mapsfromsimpleextensions.122.2.Splittingfields132.3.Algebraicclosures143.TheFundamentalTheoremofGaloisTheory183.1.Multipleroots

4、183.2.Groupsofautomorphismsoffields193.3.Separable,normal,andGaloisextensions213.4.ThefundamentaltheoremofGaloistheory233.5.Constructiblenumbersrevisited263.6.Galoisgroupofapolynomial263.7.Solvabilityofequations27Copyright1996J.S.Milne.Youmaymakeonecopyofthesenotesforyourownpersonaluse.i

5、iiJ.S.MILNE4.ComputingGaloisGroups.284.1.WhenisGf⊂An?284.2.WhenisGftransitive?294.3.Polynomialsofdegree≤3294.4.Quarticpolynomials294.5.ExamplesofpolynomialswithSpasGaloisgroupoverQ314.6.Finitefields324.7.ComputingGaloisgroupsoverQ335.ApplicationsofGaloisTheory365.1.Primitiveelementtheore

6、m.365.2.FundamentalTheoremofAlgebra385.3.Cyclotomicextensions395.4.Independenceofcharacters415.5.Hilbert’sTheorem90.425.6.Cyclicextensions.445.7.ProofofGalois’ssolvabilitytheorem455.8.Thegeneralpolynomialofdegreen46Symmetricpolynomials46Thegeneralpolynomial47Abriefhistory495.9.Normsandt

7、races495.10.InfiniteGaloisextensions(sketch)526.TranscendentalExtensions54FIELDSANDGALOISTHEORY11.ExtensionsofFields1.1.Definitions.AfieldisasetFwithtwocompositionlaws+and·suchthat(a)(F,+)isanabeliangroup;(b)letF×=F−{0};then(F×,·)isanabeliangroup;(c)(distributivelaw)foralla,b,c∈F,