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

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210 Ramification points are indicated by small white discs. The cuts are black. A dashed vertical surface indicates a connection between two nonadjacent sheets (i.e., between which there are other sheets): it always corresponds to a cut. At every cut 2 sheets meet: going along a smooth curve on the surface one moves from one sheet to another (see Figure 117). FIGURE 117 The self-intersection lines of the surface which are not cuts are white. By a transversal crossing of a line of this type along a smooth curve lying on the surface one remains on the same sheet (see Figure 118). FIGURE 118 The following Riemann surfaces are considered in Problems 298–317.
FIGURE 119 211
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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
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The complex numbers 49 where is the
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The complex numbers 51 Consequently
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The complex numbers 53 For complex
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The complex numbers 55 think that n
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The complex numbers 57 214. Let us
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The complex numbers 59 219. Prove t
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The complex numbers 61 functions (s
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The complex numbers 63 of continuit
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The complex numbers 65 erations of
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The complex numbers 67 a) b) c) d)
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The complex numbers 69 REMARK. If a
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The complex numbers 71 2.8 Images o
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The complex numbers 73 where all th
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The complex numbers 75 in such func
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The complex numbers 77 FIGURE 26 no
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The complex numbers 79 C from to th
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The complex numbers 81 DEFINITION.
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The complex numbers 83 FIGURE 32 cu
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The complex numbers 85 to infinity
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The complex numbers 87 property is
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The complex numbers 89 means will b
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The complex numbers 91 For example,
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The complex numbers 93 of the Riema
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The complex numbers 95 in Figure 38
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The complex numbers 97 permutation
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The complex numbers 99 2.13 Monodro
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The complex numbers 101 First we pr
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The complex numbers 103 and prove t
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Chapter 3 Hints, Solutions, and Ans
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Solutions 107 positive integer unde
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Solutions 109 on the left and by on
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Solutions 111 39. a) See Table 8; b
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Solutions 113 of the real numbers i
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Solutions 115 to all and therefore
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Solutions 117 is a subgroup of the
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Solutions 119 hypothesis. The group
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Solutions 121 The given group is th
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Solutions 123 of symmetries of the
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Solutions 125 i.e., there exists an
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Solutions 127 Answer. The normal su
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Solutions 129 In this way the quoti
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Solutions 131 sends every into itse
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Solutions 133 ement of the commutan
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Solutions 135 143. Let be the three
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Solutions 137 150. See 57. Suppose
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Solutions 139 subgroup. Hence if a
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Solutions 141 is commutative. Since
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Solutions 143 180. Since the lower
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Solutions 145 FIGURE 43 190. The pe
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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
Page 175 and 176: Solutions 159 function of real argu
Page 177 and 178: Solutions 161 i.e., Consider the re
Page 179 and 180: Solutions 163 Choose as the smalles
Page 181 and 182: Solutions 165 Since the function is
Page 183 and 184: Solutions 167 FIGURE 55 FIGURE 56 F
Page 185 and 186: Solutions 169 FIGURE 60 257. Suppos
Page 187 and 188: Solutions 171 265. 266. 267. If the
Page 189 and 190: Solutions 173 following way. If the
Page 191 and 192: Solutions 175 for all such that and
Page 193 and 194: Solutions 177 b) In order for to va
Page 195 and 196: Solutions 179 to the condition coin
Page 197 and 198: Solutions 181 are continuous functi
Page 199 and 200: Solutions 183 turn around the other
Page 201 and 202: Solutions 185 sought is shown in Fi
Page 203 and 204: Solutions 187 FIGURE 86 FIGURE 87 p
Page 205 and 206: Solutions 189 sum of multi-valued f
Page 207 and 208: Solutions 191 316. a) Let and be th
Page 209 and 210: Solutions 193 by the formal method,
Page 211 and 212: Solutions 195 The correct scheme is
Page 213 and 214: Solutions 197 branch of the functio
Page 215 and 216: Solutions FIGURE 110 FIGURE 111 329
Page 217 and 218: Solutions 201 It follows that and b
Page 219 and 220: Solutions 203 340. To every sheet o
Page 221 and 222: Solutions 205 Since by hypothesis t
Page 223 and 224: Solutions 207 moving along a curve
Page 225: Drawings of Riemann surfaces 209 Th
Page 229 and 230: FIGURE 121 213
Page 237 and 238: Appendix by A. Khovanskii: Solvabil
Page 239 and 240: Solvability of Equations 223 functi
Page 241 and 242: Solvability of Equations 225 Suppos
Page 243 and 244: Solvability of Equations 227 where
Page 245 and 246: Solvability of Equations 229 finds
Page 247 and 248: Solvability of Equations 231 quadra
Page 249 and 250: Solvability of Equations 233 More p
Page 251 and 252: Solvability of Equations 235 group.
Page 253 and 254: Solvability of Equations 237 Every
Page 255 and 256: Solvability of Equations 239 The gr
Page 257 and 258: Solvability of Equations 241 FIG. 1
Page 259 and 260: Solvability of Equations 243 are ob
Page 261 and 262: Solvability of Equations 245 corres
Page 265 and 266: Solvability of Equations 249 up to
Page 267 and 268: Solvability of Equations 251 fined
Page 269 and 270: Solvability of Equations 253 the qu
Page 271 and 272: Solvability of Equations 255 the no
Page 273 and 274: 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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Abel's theorem in problems and solutions - School of Mathematics

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