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# Abel's Theorem in Problems and Solutions: Based on the lectures of Professor V.I. Arnold

## byV.B. Alekseev

### Paperback | December 1, 2010

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The following ISBNs are associated with this title:

ISBN - 10:9048166098

ISBN - 13:9789048166091

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Table of Contents

Preface 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*= SQRT*z*.**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.

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