文档介绍：ABEL’S THEOREM IN PROBLEMS AND SOLUTIONS
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Abel’s Theorem
in Problems and Solutions
Based on the lectures of Professor . Arnold
by
. Alekseev
Moscow State University,
Moscow, Russia
KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
eBook ISBN: 1-4020-2187-9
Print ISBN: 1-4020-2186-0
©2004 Springer Science + Business Media, Inc.
Print ©2004 Kluwer Academic Publishers
Dordrecht
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Contents
Preface for the English edition by . Arnold ix
Preface xiii
Introduction 1
1 Groups 9
Examples 9
Groups of transformations 13
Groups 14
Cyclic groups 18
Isomorphisms 19
Subgroups 21
Direct product 23
Cosets. Lagrange’s theorem 24
Internal automorphisms 26
Normal subgroups 28
Quotient groups 29
31
Homomorphisms 33
Soluble groups 38
Permutations 40
2 plex numbers 45
Fields and polynomials 46
The field plex numbers 51
Uniqueness of the field plex
numbers 55
Geometrical descriptions of the
complex numbers 58
v
vi
The trigonometric form of plex numbers 60
Continuity 62
Continuous curves 65
Images of curves: the basic theorem
of the algebra plex numbers 71
The Riemann surface of the function 74
The Riemann surfaces of more
complicated functions 83
Functions representable by radicals 90
Monodromy groups of multi-valued
functions 96
Monodromy groups of functions
representable by radicals 99
The Abel theorem 100
3 Hints, Solutions, and Answers 105
Problems of Chapter 1 105
Problems