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

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98 Chapter 2 333. To which of the groups you already know are the permutation groups of the schemes of the functions considered in Problems: 1) 314; 2) 317; 3) 319 isomorphic? 334. For the two schemes of the function built in the solution of Problem 328, describe the permutation groups. Suppose that the point is neither a branch point nor a non-uniqueness point of the multi-valued function and that are all values of the function at the point Consider a continuous curve C starting and ending at the point and not passing through any branch point and any non-uniqueness point of the function Take a value and define by continuity along the curve C a new value Starting from distinct values we obtain different values (otherwise along the curve C the uniqueness would be lost). Hence to the curve C there corresponds a certain permutation of the values Thus if to the curve C there corresponds the permutation to the curve there corresponds the permutation and if to the curves and (both with ending points at there correspond the permutations and then to the curve there corresponds the permutation (recall that the permutations in a product are kept from right to left). In this way if we consider all possible curves starting and ending at then the corresponding permutations will form a group, the group of permutations of the values 335. Let be the permutation group of the values and the permutation group of some scheme of the function Prove that the groups and are isomorphic. Notice that in the definition of the permutation group of the values one does not use any scheme of the Riemann surface of the function From the result of Problem 335 it then follows that the permutation group of the values for an arbitrary point and the permutation group of an arbitrary scheme of the Riemann surface of the function are isomorphic. Consequently the permutation group of the values for all points and the permutation group of all the schemes of the Riemann surfaces of the function are isomorphic, i.e., they represent one unique group. This group is called the monodromy group of the multivalued function
The complex numbers 99 2.13 Monodromy groups of functions representable by radicals In this section we prove one of the main theorems of this book. THEOREM 11. If the multi-valued function is representable by radicals its monodromy group is soluble (cf., §1.14). The proof of Theorem 11 consists in the solutions of the following problems. 336. Let or or or and suppose that we have built the scheme of the Riemann surface of the function from the schemes of the Riemann surfaces of the functions and by the formal method (cf., Theorem 8 (a), §2.11). Prove that if F and G are the permutation groups of the initial schemes, then the permutation group of the scheme built is isomorphic to a subgroup of the direct product G × F (cf., Chapter 1, 1.7). 337. Let be the permutation group of the scheme built by the formal method under the hypotheses of the preceding problem, and let be the group of permutations of the real scheme of the Riemann surface of the function Prove that there exists a surjective homomorphism (cf., §1.7) of the group onto the group 338. Suppose the monodromy groups of the functions and be soluble. Prove that the monodromy groups of the functions soluble as well. 339. Suppose the monodromy group of the function be soluble. Prove that the monodromy group of the function is also soluble. 340. Let H be the permutation group of a scheme of the function and F the permutation group of a scheme of the function made with the same cuts. Define a surjective homomorphism of the group H onto the group F. 341. Prove that the kernel of the homomorphism (cf., §1.13) defined by the solution of the preceding problem is commutative. are
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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
Page 63 and 64: 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: The complex numbers 97 permutation
Page 117 and 118: The complex numbers 101 First we pr
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
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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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