国立交通大学应用数学系 博士班资格考命题参考书籍与范.pdf

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1、國立交通大學應用數學系博士班資格考命題參考書籍與範圍96年11月修定(97學年度入學新生適用)一、代數參考書籍:[1]JohnB.Fraleigh,AFirstCourseinAbstractAlgebra,PearsonEducation,2003.[2]I.N.Herstein,AbstractAlgebra,Prentice-Hall,1996.[3]SergeLang,Algebra,GraduateTextsinMathematics211,Springer,2002.(Sections1.1-1.8,2.1-2.5,3.1-3.7,5.

2、1-5.5,6.1,6.2).範圍I.Groups([1,Chapters1,2,3,7],[2,Chapters2,3],[3,Sections1.1-1.8])GroupsandSubgroupsCyclicgroupsGroupsofPermutationsCosetsandLagrange’sTheoremDirectProductsandFiniteAbelianGroupsHomomorphismsFactorGroupsGroupActiononaSetIsomorphismTheoremsSeriesofGroupsSylow’sT

3、heoremsII.Rings([1,Chapters4,5,6],[2,Chapter4],[3,Chapter2])RingsandFieldsIntegralDomainsFermat’sandEuler’sTheoremsRingsofPolynomialsFactorizationofPolynomialsoveraFieldHomomorphismsandFactorRingsPrimeandMaximalIdealsUniqueFactorizationDomainsEuclideanDomainsGaussianIntegers1I

4、II.Modules([3,Sections3.1-3.7])DirectProductsandSumsofModulesFreeModulesVectorSpacesDualSpaceandDualModuleModulesoverPrincipalRingsIV.Fields([1,Chapters6,10],[2,Chapters5,6],[3,Sections5.1-5.5,6.1,6.2])ExtensionFieldsandVectorSpacesAlgebraicExtensionsRootsofPolynomialsFiniteFi

5、eldsCyclotomicPolynomialsAutomorphismsandGaloisTheory2二、幾何參考書[1]ManfredoP.doCarmo,DifferentialGeometryofCurvesandSurfaces,Prentice-Hall1976.[2]I.M.Singer,J.A.Thorpe,LectureNotesonElementaryTopologyandGeometry,Springer-Verlag1976.範圍Chapter1of[1]:Frenetformula,thelocalcanonicalf

6、orm.Chapters2,3and4of[1]:Gaussmap,geodesics,orientation,Gauss-BonnetTheorem.Chapter5of[1]:Completesurfaces,Jacobifields,conjugatepoints.Chapter3of[2]:Fundamentalgroup,coveringspaces.Chapters5and6of[2]:Manifolds,differentialforms,Stokestheorem,DeRhamTheorem.3三、分析參考書rd[1]H.L.Roy

7、den,Realanalysis,3ed.,1989.[2]A.Friedman,Foundationsofmodernanalysis,1970,Chapters1-4.範圍1.Lebesguemeasureandintegral:Measurablefunction,Littlewood’sthreeprinciples,Fatou’slemma,monotoneconvergencetheorem,dominatedconvergencetheorem.2.Differentiationandintegration:Functionofbou

8、ndedvariation,absolutecontinuousfunction,singularfunction,fun