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二十面体和5次方程的解的讲义（英文版）

样章
作者：
Felix Klein
定价：
168.00元
ISBN：
978-7-04-051022-5
版面字数：
420.000千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2019-05-27
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
代数几何学
暂无
  • 前辅文
  • Part I Theory of the Icosahedron Itself
    • Chapter I The Regular Solids and the Theory of Groups
      • 1.Statement of the Question.
      • 2.Preliminary Notions of the Group-Theory
      • 3.The Cyclic Rotation Groups
      • 4.The Group of theDihedral Rotations.
      • 5.The Quadratic Group.
      • 6.The Group of the Tetrahedral Rotations.
      • 7.The Group of the Octahedral Rotations
      • 8.The Group of the Icosahedral Rotations
      • 9.On the Planes of Symmetry inOur Configurations
      • 10.General Groups of Points—Fundamental Domains
      • 11.The Extended Groups
      • 12.Generation of the IcosahedralGroup.
      • 13.Generation of the Other Groups of Rotations.
    • Chapter II Introduction of (x +i y)
      • 1.First Presentation and Survey of the Developments of This Chapter
      • 2.On Those Linear Transformations of (x +i y) WhichCorrespond to Rotations Round the Centre
      • 3.Homogeneous Linear Substitutions—Their Composition.
      • 4.Return to the Groups of Substitutions—the Cyclic and DihedralGroups
      • 5.The Groups of the Tetrahedron andOctahedron
      • 6.The IcosahedralGroup.
      • 7.Non-Homogeneous Substitutions—Consideration of the Extended Groups.
      • 8.Simple Isomorphism in the Case of Homogeneous Groups of Substitutions
      • 9.Invariant Forms Belonging to a Group—The Set of Forms for the Cyclic andDihedralGroups
      • 10.Preparation for the Tetrahedral andOctahedral Forms.
      • 11.The Set of Forms for the Tetrahedron.
      • 12.The Set of Forms for the Octahedron
      • 13.The Set of Forms for the Icosahedron.
      • 14.The Fundamental Rational Functions
      • 15.Remarks on the Extended Groups
    • Chapter III Statement and Discussion of the Fundamental Problem, According to the Theory of Functions
      • 1.Definition of the Fundamental Problem.
      • 2.Reduction of the Form-Problem.
      • 3.Plan of the Following Investigations
      • 4.On the Conformable Representation byMeans of the Function z(Z)
      • 5.March of the z1, z2 Function in General—Development in Series
      • 6.Transition to theDifferential Equations of the ThirdOrder.
      • 7.Connection with Linear Differential Equations of the Second Order
      • 8.Actual Establishment of the Differential Equation of the Third Order for z[Z].
      • 9.Linear Differential Equations of the Second Order for z1 and z2
      • 10.Relations to Riemann’s P-Function.
    • Chapter IV On the Algebraical Character of Our Fundamental Problem
      • 1.Problemof the Present Chapter
      • 2.On the Group of an Algebraical Equation.
      • 3.General Remarks on Resolvents.
      • 4.The Galois Resolvent in Particular
      • 5.Marshalling of our Fundamental Equations
      • 6.Consideration of the Form-Problems
      • 7.The Solution of the Equations of the Dihedron,Tetrahedron,andOctahedron.
      • 8.The Resolvents of the Fifth Degree for the Icosahedral Equation
      • 9.The Resolvent of the r ’s
      • 10.Computation of the Forms t andW
      • 11.The Resolvent of the u’s.
      • 12.The Canonical Resolvent of the Y ’s
      • 13.Connection of the New Resolvent with the Resolvent of the r ’s
      • 14.On the Products of Differences for the u’s and the Y ’s.
      • 15.The Simplest Resolvent of the SixthDegree.
      • 16.Concluding Remarks.
    • Chapter V General Theorems and Survey of the Subject.
      • 1.Estimation of our Process of Thought so far, and Generalisations Thereof
      • 2.Determination of all Finite Groups of Linear Substitutions of a Variable
      • 3.Algebraically Integrable Linear Homogeneous Differential Equations of the Second Order
      • 4.Finite Groups of Linear Substitutions for a Greater Number of Variables.
      • 5.Preliminary Glance at the Theory of Equations of the Fifth Degree,and Formulation of a General Algebraical Problem.
      • 6.InfiniteGroups of Linear Substitutions of a Variable
      • 7.Solution of the Tetrahedral, Octahedral, and Icosahedral Equations by EllipticModular Functions.
      • 8.Formulae for the Direct Solution of the Simplest Resolvent of the SixthDegree for the Icosahedron
      • 9.Significance of the Transcendental Solutions.
  • Part II Theory of Equations of the Fifth Degree
    • Chapter I The HistoricalDevelopment of the Theory of Equations of the Fifth Degree
      • 1.Definition of Our First Problem.
      • 2.Elementary Remarks on the Tschirnhausian Transformation—Bring’s Form.
      • 3.Data Concerning Elliptic Functions
      • 4.On Hermite’sWork of 1858
      • 5.The Jacobian Equations of the SixthDegree
      • 6.Kronecker’sMethod for the Solution of Equations of the Fifth Degree
      • 7.On Kronecker’sWork of 1861.
      • 8.Object of our Further Developments
    • Chapter II Introduction of GeometricalMaterial
      • 1.Foundation of the Geometrical Interpretation
      • 2.Classification of the Curves and Surfaces.
      • 3.The Simplest Special Cases of Equations of the Fifth Degree.
      • 4.Equations of the Fifth DegreeWhich Appertain to the Icosahedron.
      • 5.Geometrical Conception of the Tschirnhausian Transformation.
      • 6.Special Applications of the Tschirnhausian Transformation
      • 7.Geometrical Aspect of the Formation of Resolvents
      • 8.On Line Co-ordinates in Space.
      • 9.A Resolvent of the Twentieth Degree of Equations of the Fifth Degree
      • 10.Theory of the Surface of the Second Degree.
    • Chapter III The Canonical Equations of the Fifth Degree
      • 1.Notation–The Fundamental Lemma.
      • 2.Determination of the Appropriate Parameter λ
      • 3.Determination of the Parameter μ.
      • 4.The Canonical Resolvent of the Icosahedral Equation.
      • 5.Solution of the Canonical Equations of the Fifth Degree
      • 6.Gordan’s Process
      • 7.Substitutions of the λ,μ’s—Invariant Forms.
      • 8.General Remarks on the Calculations WhichWe Have to Perform.
      • 9.Fresh Calculation of theMagnitude m1
      • 10.Geometrical Interpretation ofGordan’s Theory
      • 11.Algebraical Aspects (After Gordan)
      • 12.The Normal Equation of The rν’s
      • 13.Bring’s Transformation
      • 14.TheNormal Equation ofHermite.
    • Chapter IV The Problem of the A’s and the Jacobian Equations of the Sixth Degree
      • 1.The Object of the Following Developments
      • 2.The Substitutions of the A’s—Invariant Forms.
      • 3.Geometrical Interpretation—Regulation of the Invariant Expressions
      • 4.The Problem of the A’s and Its Reduction.
      • 5.On the Simplest Resolvents of the Problem of the A’s
      • 6.The General Jacobian Equation of the SixthDegree
      • 7.Brioschi’s Resolvent
      • 8.Preliminary Remarks on the Rational Transformation of Our Problem.
      • 9.Accomplishment of the Rational Transformation
      • 10.Group-Theory Significance of Cogredience and Contragredience
      • 11.Introductory to the Solution of Our Problem.
      • 12.Corresponding Formulae.
    • Chapter V The General Equation of the Fifth Degree
      • 1.Formulation of TwoMethods of Solution.
      • 2.Accomplishment of Our FirstMethod
      • 3.Criticismof theMethods of Bring andHermite.
      • 4.Preparation for Our SecondMethod of Solution.
      • 5.Of the Substitutions of the A,A’s—Definite Formulation
      • 6.The Formulae of Inversion ofOur SecondMethod
      • 7.Relations to Kronecker and Brioschi
      • 8.Comparison of Our TwoMethods.
      • 9.On theNecessity of the Accessory Square Root
      • 10.Special Equations of the Fifth DegreeWhich Can Be Rationally Reduced to an Icosahedral Equation.
      • 11.Kronecker’s Theorem.
  • Appendix

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