Abels Theorem in Problems and Solutions
ISBN 9781402021862
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InhaltsangabePreface for the English edition; V.I. Arnold. Preface. Introduction.
1: Groups. 1.1. Examples. 1.2. Groups of transformations. 1.3. Groups. 1.4. Cyclic groups. 1.5. Isomorphisms. 1.6. Subgroups. 1.7. Direct product. 1.8. Cosets. Lagrange's theory. 1.9. Internal automorphisms. 1.10. Normal subgroups. 1.11. Quotient groups. 1.12. Commutant. 1.13. Homomorphisms. 1.14. Soluble groups. 1.15. Permutations.
2: The complex numbers. 2.1. Fields and polynomials. 2.2. The field of complex numbers. 2.3. Uniqueness of the field of complex numbers. 2.4. Geometrical descriptions of the field of complex numbers. 2.5. The trigonometric form of the complex numbers. 2.6. Continuity. 2.7. Continuous curves. 2.8. Images of curves: the basic theorem of the algebra of complex numbers. 2.9. The Riemann surface of the function w = SQRTz. 2.10. The Riemann surfaces of more complicated functions. 2.11. Functions representable by radicals. 2.12. Monodromy groups of multi-valued functions. 2.13. Monodromy groups of functions representable by radicals. 2.14. The Abel theorem.
3: Hints, Solutions and Answers. 3.1.Problems of Chapter 1. 3.2. Problems of Chapter 2. Drawings of Riemann surfaces; F. Aicardi.
Appendix. Solvability of equations by explicit formulae; A. Khovanskii. A.1. Explicit solvability of equations. A.2. Liouville's theory. A.3. Picard-Vessiot's theory. A.4. Topological obstructions for the representation of functions by quadratures. A.5. S-functions. A.6. Monodromy group. A.7. Obstructions for the representability of functions by quadratures. A.8. Solvability of algebraic equations. A.9. The monodromy pair. A.10. Mapping of the semi-plane to a polygon bounded by arcs of circles. A.11. Topological obstructions for the solvability of differential equations. A.12. Algebraic functions of several variables. A.13. Functions of several complex variables representable by quadratures and generalized quadratures. A.14. SC-germs. A.15. Topological obstruction for the solvability of the holonomic systems of linear differential equations. A.16. Topological obstruction for the solvability of the holonomic systems of linear differential equations. Bibliography.
Appendix; V.I. Arnold.
Index.
1: Groups. 1.1. Examples. 1.2. Groups of transformations. 1.3. Groups. 1.4. Cyclic groups. 1.5. Isomorphisms. 1.6. Subgroups. 1.7. Direct product. 1.8. Cosets. Lagrange's theory. 1.9. Internal automorphisms. 1.10. Normal subgroups. 1.11. Quotient groups. 1.12. Commutant. 1.13. Homomorphisms. 1.14. Soluble groups. 1.15. Permutations.
2: The complex numbers. 2.1. Fields and polynomials. 2.2. The field of complex numbers. 2.3. Uniqueness of the field of complex numbers. 2.4. Geometrical descriptions of the field of complex numbers. 2.5. The trigonometric form of the complex numbers. 2.6. Continuity. 2.7. Continuous curves. 2.8. Images of curves: the basic theorem of the algebra of complex numbers. 2.9. The Riemann surface of the function w = SQRTz. 2.10. The Riemann surfaces of more complicated functions. 2.11. Functions representable by radicals. 2.12. Monodromy groups of multi-valued functions. 2.13. Monodromy groups of functions representable by radicals. 2.14. The Abel theorem.
3: Hints, Solutions and Answers. 3.1.Problems of Chapter 1. 3.2. Problems of Chapter 2. Drawings of Riemann surfaces; F. Aicardi.
Appendix. Solvability of equations by explicit formulae; A. Khovanskii. A.1. Explicit solvability of equations. A.2. Liouville's theory. A.3. Picard-Vessiot's theory. A.4. Topological obstructions for the representation of functions by quadratures. A.5. S-functions. A.6. Monodromy group. A.7. Obstructions for the representability of functions by quadratures. A.8. Solvability of algebraic equations. A.9. The monodromy pair. A.10. Mapping of the semi-plane to a polygon bounded by arcs of circles. A.11. Topological obstructions for the solvability of differential equations. A.12. Algebraic functions of several variables. A.13. Functions of several complex variables representable by quadratures and generalized quadratures. A.14. SC-germs. A.15. Topological obstruction for the solvability of the holonomic systems of linear differential equations. A.16. Topological obstruction for the solvability of the holonomic systems of linear differential equations. Bibliography.
Appendix; V.I. Arnold.
Index.