Abel's Theorem in Problems and Solutions - V.B. Alekseev

Abel's Theorem in Problems and Solutions - V.B. Alekseev -...

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ABEL’S THEOREM IN PROBLEMS AND SOLUTIONS
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Abel’s Theorem in Problems and Solutions Based on the lectures of Professor V.I. Arnold by V.B. Alekseev Moscow State University, Moscow, Russia KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
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eBook ISBN: 1 - 4020 - 2187 - 9 Print ISBN: 1 - 4020 - 2186 - 0 ©2004 Springer Science + Business Media, Inc. Print © 2004 Kluwer Academic Publishers All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: http://www.ebooks.kluweronline.com and the Springer Global Website Online at: http://www.springeronline.com Dordrecht
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Contents Preface for the English edition by V.I. Arnold Preface Introduction 1 Groups 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 Examples Groups of transformations Groups Cyclic groups Isomorphisms Subgroups Direct product Cosets. Lagrange’s theorem Internal automorphisms Normal subgroups Quotient groups Commutant Homomorphisms Soluble groups Permutations ix xiii 1 9 9 13 14 18 19 21 23 24 26 28 29 31 33 38 40 2.1 2.2 2.3 2.4 Fields and polynomials The field of complex numbers Uniqueness of the field of complex numbers Geometrical descriptions of the complex numbers 46 51 55 2 The complex numbers 45 v 58
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vi 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 The trigonometric form of the complex numbers Continuity Continuous curves Images of curves: the basic theorem of the algebra of complex numbers The Riemann surface of the function The Riemann surfaces of more complicated functions Functions representable by radicals Monodromy groups of multi-valued functions Monodromy groups of functions representable by radicals The Abel theorem 60 62 65 71 74 83 90 96 99 100 3 Hints, Solutions, and Answers 3.1 3.2 Problems of Chapter 1 Problems of Chapter 2 Drawings of Riemann surfaces (F. Aicardi) 105 148 209 Appendix by A. Khovanskii: Solvability of equations by explicit formulae A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.10.1 A.10.2 Explicit solvability of equations Liouville’s theory Picard–Vessiot’s theory Topological obstructions for the representation of functions by quadratures Monodromy group Obstructions for the representability of functions by quadratures Solvability of algebraic equations The monodromy pair Mapping of the semi-plane to a polygon bounded by arcs of circles Application of the symmetry principle Almost soluble groups of homographic and conformal mappings 221 222 224 228 230 231 232 233 234 235 237 238
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A.10.3 The integrable case A.11 Topological obstructions for the solvability of differential equations A.11.1 The monodromy group of a linear differential equation and its relation with the Galois group A.11.2 Systems of differential equations of Fuchs’ type with small coefficients A.12 Algebraic functions of several variables A.13 Functions of several complex variables representable by quadratures and generalized quadratures A.14 A.15 Topological obstructions for the representability by quadratures
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Abel's Theorem in Problems and Solutions - V.B. Alekseev -...

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