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

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102 Chapter 2 Riemann surface of the function at the point (more precisely, along the cuts joining the point to infinity; cf., Remark 2, §2.10) join? 347. Let be a function expressing the roots of equation (2.8) in terms of the parameter Moreover, let and be two arbitrary points different from and and and be two arbitrary images of these points under the mapping Prove that it is possible to draw a continuous curve joining the points and not passing through the points and and such that its continuous image, starting from the point ends at the point 348. Prove that all four points and are branch points of the function How can we represent the scheme of the Riemann surface of the function Draw all different possible schemes (we consider different two schemes if they cannot be obtained one from another by a permutation of the sheets and of the branch points). 349. Find the monodromy group of the function expressing the roots of the equation in terms of the parameter 350. Prove that the function expressing the roots of the equation in terms of the parameter is not representable by radicals. 351. Prove that the algebraic general equation of fifth degree (where are complex parameters, is not solvable by radicals, i.e., that there are no formulae expressing the roots of this equation in terms of the coefficients by means of the operations of addition, subtraction, multiplication, division, elevation to an integer power and extraction of a root of integer order. 352. Considerer the equation
The complex numbers 103 and prove that for the general algebraic equation of degree is not solvable by radicals. The results of Problems 351 and 352 contain the proof of the theorem which is the subject of the present book. We have indeed proved the ABEL’S THEOREM. For the general algebraic equation of degree is not solvable by radicals. REMARK 1. In the introduction we deduced the Cardano formulae for the solution of the general algebraic equation of third degree. The roots of the equation are not given by all values expressed by these formulae, but only by those which satisfy some supplementary conditions. One may therefore pose the question of whether it is possible, also for the general equation of degree to build by radicals a formula such that the roots of the equation are only a part of the values that are expressed by this formula. This is not possible even for equation (2.8). Indeed, if the values of the function expressing the roots of equation (2.8) in terms of the parameter are only a part of the values of a function represented by radicals, then the Riemann surface of the function is a separate part of the Riemann surface of the function If G is the monodromy group of the function then to every permutation of the group G there corresponds a permutation of the five sheets of the function This mapping is a homomorphism of the group G into the group Since the group is not soluble then the group G is also not soluble (cf., 163). On the other hand, the group G must be soluble, being the monodromy group of a function representable by radicals. We have thus obtained a contradiction. REMARK 2. From Remark 1 in §2.13 it follows that the Abel theorem also holds if one is allowed to use, besides radicals, some other functions, for example all analytic functions (such as etc.), the function ln and some others. REMARK 3. Consider equation (2.8) only in the domain of real numbers. Suppose that the function expresses the real roots of the equation
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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
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: The complex numbers 101 First we pr
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
Page 167 and 168: 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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