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

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xiv otherwise. He will learn what a Riemann surface is and of what the ‘basic theorem of the complex numbers algebra’ consists. The author will accompany the reader along this path, but he will also give him the possibility of testing his own forces. For this purpose he will propose to the reader a large number of problems. The problems are posed directly within the text, so representing an essential part of it. The problems are labelled by increasing numbers in bold figures. Whenever the problem might be too difficult for the reader, the chapter ‘Hint, Solutions, and Answers’ will help him. The book contains many notions which may be new to the reader. To help him in orienting himself amongst these new notions we put at the end of the book an alphabetic index of notions, indicating the pages where their definitions are to be found. The proof of the Abel theorem presented in this book was presented by professor Vladimir Igorevich Arnold during his lectures to the students of the la 11th course of the physics-mathematics school of the State University of Moscow in the years 1963–64. The author of this book, who at that time was one of the pupils of that class, during the years 1970– 71 organized for the pupils of that school a special seminar dedicated to the proof of the Abel theorem. This book consists of the material collected during these activities. The author is very grateful to V.I. Arnold for having made a series of important remarks during the editing of the manuscript. V.B. Alekseev
Introduction We begin this book by examining the problem of solving algebraic equations in one variable from the first to the fourth degree. Methods for solving equations of first and second degree were already known by the ancient mathematicians, whereas the methods of solution of algebraic equations of third and fourth degree were invented only in the XVI century. An equation of the type: in which 2 , is called the generic algebraic equation of degree in one variable. For we obtain the linear equation This equation has the unique solution for any value of the coefficients. For we obtain the quadratic equation (in place of we write as learnt in school). Dividing both members of this equation by and putting and we obtain the reduced equation 2 For the time being the coefficients may be considered to be arbitrary real numbers. 1
Page 3 and 4: ABEL’S THEOREM IN PROBLEMS AND SO
Page 5 and 6: Abel’s Theorem in Problems and So
Page 7 and 8: Contents Preface for the English ed
Page 9 and 10: A.10.3 The integrable case A.11 Top
Page 11 and 12: Preface to the English edition by V
Page 13 and 14: of the differential equations. Unfo
Page 15: Preface In high school algebraic eq
Page 19 and 20: Introduction 3 where by is indicate
Page 21 and 22: Introduction 5 By Viète’s theore
Page 23 and 24: Introduction 7 We will be able to p
Page 25 and 26: Chapter 1 Groups 1.1 Examples In ar
Page 27 and 28: Groups 11 the triangle ABC into its
Page 29 and 30: Groups 13 7. Find all symmetries of
Page 31 and 32: Groups 15 notations (see the soluti
Page 33 and 34: Groups 17 element, which we will de
Page 35 and 36: Groups 19 35. Suppose that is the m
Page 37 and 38: Groups 21 If a group is isomorphic
Page 39 and 40: Groups 23 tude); (rotation by 180°
Page 41 and 42: Groups 25 Hence left cosets generat
Page 43 and 44: Groups 27 as a permutation of the t
Page 45 and 46: Groups 29 102. Let be the order of
Page 47 and 48: Groups 31 111. Find all normal subg
Page 49 and 50: Groups 33 FIGURE 8 127. Prove that
Page 51 and 52: Groups 35 137. Prove that ker is a
Page 53 and 54: Groups 37 We now observe what happe
Page 55 and 56: Groups 39 FIGURE 12 vertices; 4) ro
Page 57 and 58: Groups 41 Every permutation of degr
Page 59 and 60: Groups 43 180. Prove that by multip
Page 61 and 62: 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 and 114:
The complex numbers 97 permutation
Page 115 and 116:
The complex numbers 99 2.13 Monodro
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
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Solutions 121 The given group is th
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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
Page 167 and 168:
Solutions 151 necessarily that and
Page 169 and 170:
Solutions 153 Let C be the field of
Page 171 and 172:
Solutions 155 in as numerators and
Page 173 and 174:
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 and 226:
Drawings of Riemann surfaces 209 Th
Page 227 and 228:
FIGURE 119 211
Page 229 and 230:
FIGURE 121 213
Page 231 and 232:
FIGURE 123 215
Page 233 and 234:
FIGURE 125 217
Page 235 and 236:
FIGURE 127 219
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 263 and 264:
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
Page 275 and 276:
Page 277 and 278:
Bibliography [1] [2] [3] [4] [5] [6
Page 279 and 280:
[23] [24] [25] [26] [27] [28] 263 V
Page 281 and 282:
Appendix (V.I. Arnold) The topologi
Page 283 and 284:
Index Abel’s theorem, 6, 103 addi
Page 285:
pack of sheets, 96 parametric equat
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