Abel's theorem in problems and solutions - School of Mathematics

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100 Chapter 2 342. Suppose the monodromy group F of the function be soluble. Prove that the monodromy group H of the function is also soluble. The functions and are continuous single-valued functions on the entire plane. Their Riemann surfaces thus consist of a single sheet and therefore the corresponding monodromy groups consist of a single element and are therefore soluble. As a consequence, taking into account the definition of the functions representable by radicals (§2.11) and the results of Problems 338, 339, 342, one obtains the proposition of Theorem 11. REMARK 1. This remark is for that reader who knows the theory of analytic functions. Theorem 11 holds for a wider class of functions. For example, to define a function one can be allowed to use, besides of constant functions, the identity function, the functions expressed by arithmetic operations and radicals, also all analytic single-valued functions (for example, etc.), the multi-valued function ln and some others. In this case the monodromy group of the function will be soluble, though it is not necessarily finite. 2.14 The Abel theorem Consider the equation We consider as a parameter and for every complex value of we look for all complex roots of this equation. By virtue of the result of Problem 269 the given equation for every has 5 roots (taking into account the multiplicities). 343. Which values of can be multiple roots (of order higher than 1, cf., §2.8) of the equation For which values of are these roots multiples? It follows from the solution of the preceding problem that for and equation (2.8) has four distinct roots, and for the other values of it has 5 distinct roots. Let us study the function
The complex numbers 101 First we prove that for small variations of the parameter the roots of equation (2.8) vary only slightly. This property is more precisely expressed by the following problem. 344. Let be an arbitrary complex number and be one of the roots of equation (2.8) for Consider a disc of radius arbitrarily small with its centre at Prove that there exists a real number such that if then in the disc considered there exists at least one root of equation (2.8) for also. Suppose the function express the roots of equation (2.8) in terms of the parameter and be one of the values It follows from the result of Problem 344 that if changes continuously along a curve, starting at the point then one can choose one of the values in such a way that the point too, moves continuously along a curve starting from the point In other words, the function can be defined by continuity along an arbitrary curve C. Therefore if the curve C avoids the branch points and the non-uniqueness points of the function the function is uniquely defined by continuity along the curve C. 345. Prove that points different from and can be neither branch points nor non-uniqueness points of a function expressing the roots of equation (2.8) in terms of the parameter Let be the function expressing the roots of equation (2.8) in terms of the parameter The function being an algebraic function 20 , is ‘sufficiently good’ (cf., §2.10), i.e., it possesses the monodromy property. One can therefore build for the function the Riemann surface (cf., 309 and 310). This Riemann surface evidently has 5 sheets. By virtue of the result of Problem 345 the only possible branch points of the function are the points and but it is not yet clear whether this is really the case. 346. Suppose it is known that the point (or or is a branch point of the function expressing the roots of equation (2.8) in terms of the parameter How do the sheets of the 20 The multi-valued function is said to be algebraic if it expresses in terms of the parameter all the roots of some equation in which all the are polynomials in All algebraic functions are analytic.
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ABEL’S THEOREM IN PROBLEMS AND SO
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Abel’s Theorem in Problems and So
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Contents Preface for the English ed
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A.10.3 The integrable case A.11 Top
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Preface to the English edition by V
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of the differential equations. Unfo
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Preface In high school algebraic eq
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Introduction We begin this book by
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Introduction 3 where by is indicate
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Introduction 5 By Viète’s theore
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Introduction 7 We will be able to p
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Chapter 1 Groups 1.1 Examples In ar
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Groups 11 the triangle ABC into its
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Groups 13 7. Find all symmetries of
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Groups 15 notations (see the soluti
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Groups 17 element, which we will de
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Groups 19 35. Suppose that is the m
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Groups 21 If a group is isomorphic
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Groups 23 tude); (rotation by 180°
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Groups 25 Hence left cosets generat
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Groups 27 as a permutation of the t
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Groups 29 102. Let be the order of
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Groups 31 111. Find all normal subg
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Groups 33 FIGURE 8 127. Prove that
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Groups 35 137. Prove that ker is a
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Groups 37 We now observe what happe
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Groups 39 FIGURE 12 vertices; 4) ro
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Groups 41 Every permutation of degr
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Groups 43 180. Prove that by multip
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Chapter 2 The complex numbers When
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The complex numbers 47 195. Prove t
Page 65 and 66: The complex numbers 49 where is the
Page 67 and 68: The complex numbers 51 Consequently
Page 69 and 70: The complex numbers 53 For complex
Page 71 and 72: The complex numbers 55 think that n
Page 73 and 74: The complex numbers 57 214. Let us
Page 75 and 76: The complex numbers 59 219. Prove t
Page 77 and 78: The complex numbers 61 functions (s
Page 79 and 80: The complex numbers 63 of continuit
Page 81 and 82: The complex numbers 65 erations of
Page 83 and 84: The complex numbers 67 a) b) c) d)
Page 85 and 86: The complex numbers 69 REMARK. If a
Page 87 and 88: The complex numbers 71 2.8 Images o
Page 89 and 90: The complex numbers 73 where all th
Page 91 and 92: The complex numbers 75 in such func
Page 93 and 94: The complex numbers 77 FIGURE 26 no
Page 95 and 96: The complex numbers 79 C from to th
Page 97 and 98: The complex numbers 81 DEFINITION.
Page 99 and 100: The complex numbers 83 FIGURE 32 cu
Page 101 and 102: The complex numbers 85 to infinity
Page 103 and 104: The complex numbers 87 property is
Page 105 and 106: The complex numbers 89 means will b
Page 107 and 108: The complex numbers 91 For example,
Page 109 and 110: The complex numbers 93 of the Riema
Page 111 and 112: The complex numbers 95 in Figure 38
Page 113 and 114: The complex numbers 97 permutation
Page 115: The complex numbers 99 2.13 Monodro
Page 119 and 120: The complex numbers 103 and prove t
Page 121 and 122: Chapter 3 Hints, Solutions, and Ans
Page 123 and 124: Solutions 107 positive integer unde
Page 125 and 126: Solutions 109 on the left and by on
Page 127 and 128: Solutions 111 39. a) See Table 8; b
Page 129 and 130: Solutions 113 of the real numbers i
Page 131 and 132: Solutions 115 to all and therefore
Page 133 and 134: Solutions 117 is a subgroup of the
Page 135 and 136: Solutions 119 hypothesis. The group
Page 137 and 138: Solutions 121 The given group is th
Page 139 and 140: Solutions 123 of symmetries of the
Page 141 and 142: Solutions 125 i.e., there exists an
Page 143 and 144: Solutions 127 Answer. The normal su
Page 145 and 146: Solutions 129 In this way the quoti
Page 147 and 148: Solutions 131 sends every into itse
Page 149 and 150: Solutions 133 ement of the commutan
Page 151 and 152: Solutions 135 143. Let be the three
Page 153 and 154: Solutions 137 150. See 57. Suppose
Page 155 and 156: Solutions 139 subgroup. Hence if a
Page 157 and 158: Solutions 141 is commutative. Since
Page 159 and 160: Solutions 143 180. Since the lower
Page 161 and 162: Solutions 145 FIGURE 43 190. The pe
Page 163 and 164: Solutions 147 permutations of type
Page 165 and 166: Solutions 149 is a field. 195. We h
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Solutions 151 necessarily that and
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Solutions 153 Let C be the field of
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Solutions 155 in as numerators and
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Solutions 157 c) d) e) where 226. W
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Solutions 159 function of real argu
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Solutions 161 i.e., Consider the re
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Solutions 163 Choose as the smalles
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Solutions 165 Since the function is
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Solutions 167 FIGURE 55 FIGURE 56 F
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Solutions 169 FIGURE 60 257. Suppos
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Solutions 171 265. 266. 267. If the
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Solutions 173 following way. If the
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Solutions 175 for all such that and
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Solutions 177 b) In order for to va
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Solutions 179 to the condition coin
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Solutions 181 are continuous functi
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Solutions 183 turn around the other
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Solutions 185 sought is shown in Fi
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Solutions 187 FIGURE 86 FIGURE 87 p
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Solutions 189 sum of multi-valued f
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Solutions 191 316. a) Let and be th
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Solutions 193 by the formal method,
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Solutions 195 The correct scheme is
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Solutions 197 branch of the functio
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Solutions FIGURE 110 FIGURE 111 329
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Solutions 201 It follows that and b
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Solutions 203 340. To every sheet o
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Solutions 205 Since by hypothesis t
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Solutions 207 moving along a curve
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Drawings of Riemann surfaces 209 Th
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FIGURE 119 211
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FIGURE 121 213
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FIGURE 123 215
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FIGURE 125 217
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FIGURE 127 219
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Appendix by A. Khovanskii: Solvabil
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Solvability of Equations 223 functi
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Solvability of Equations 225 Suppos
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Solvability of Equations 227 where
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Solvability of Equations 229 finds
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Solvability of Equations 231 quadra
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Solvability of Equations 233 More p
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Solvability of Equations 235 group.
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Solvability of Equations 237 Every
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Solvability of Equations 239 The gr
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Solvability of Equations 241 FIG. 1
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Solvability of Equations 243 are ob
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Solvability of Equations 245 corres
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Solvability of Equations 249 up to
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Solvability of Equations 251 fined
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Solvability of Equations 253 the qu
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Solvability of Equations 255 the no
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Solvability of Equations 257 THEORE
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Bibliography [1] [2] [3] [4] [5] [6
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[23] [24] [25] [26] [27] [28] 263 V
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Appendix (V.I. Arnold) The topologi
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Index Abel’s theorem, 6, 103 addi
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pack of sheets, 96 parametric equat
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