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

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74 Chapter 2 polynomials of first degree and those of second degree with no real roots. We had used this property in §2.3 of this chapter. Over the field of complex numbers, according to the result of Problem 269, only polynomials of first degree are irreducible. DEFINITION. Let be a root of the equation One says that is a root of order if the polynomial is divisible by and not by 273. Which is the order of the roots and in the equation DEFINITION. One calls the derivative of the polynomial the polynomial The derivative is usually denoted by the (prime). 274. Let and be two polynomials. Prove that: a) where is an arbitrary complex constant; c) 275. Let integer). Prove that 276. Prove that if the equation has a root of order then the equation has a root of order and that if the equation has a root of first order then 2.9 The Riemann surface of the function We had considered single-valued functions for which there corresponds a unique value of the function to every value of the independent variable. In what follows we will deal mainly with multi-valued functions, for which there correspond distinct values of the function to a value of the independent variable 11 . We will explain the reason for our interest 11 Whenever the context is sufficiently clear the term multi-valued will be omitted.
The complex numbers 75 in such functions. In fact, the final aim of our study is the proof of the Abel theorem, according to which a function, expressing the roots of the general equation of fifth degree, cannot be represented by radicals. But this function is multi-valued, because an equation of fifth degree has, in general, for given coefficients, five roots. Also the functions which are represented by radicals are multi-valued. The principal idea of the demonstration of the Abel theorem is the following. We put into correspondence with a multi-valued function of a complex variable a certain group, the so called monodromy group. The monodromy group of the function expressing the roots of the general equation of fifth degree in terms of a parameter cannot coincide with any monodromy group of functions representable by radicals, and therefore this function cannot be represented by radicals. In order to introduce the notion of the monodromy group we consider first another notion very important in the theory of functions of one complex variable — the notion of the Riemann 12 surface of a function. We begin by the construction of the Riemann surface for the simplest example of a multi-valued function, the function We already know that the function takes the single value for and two values for all values (cf., 229). Moreover, if is one of the values of then the other value of is 277. Find all values of: a) b) c) d) (here is the positive value of the root). Let us cut the plane along the negative side of the real axis from 0 to and for every not belonging to the cut let us choose the value which lies on the right half plane. In this way we obtain a continuous single-valued function over the whole plane, except the cut. This function, which we denote by defines a continuous and singlevalued mapping of the plane, except the cut, on the right half plane (Figure 24). REMARK. If we choose arg in such a way that then for the function we obtain (cf., 229). Therefore under the mapping the plane shrinks like a fan whose radii are shortened as its opening angle is halved. If we now choose, for every not lying on the cut, the value of which lies on the left half plane we obtain another function, still single- 12 B. Riemann (1826–1866), German mathematician.
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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
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: The complex numbers 73 where all th
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
Page 137 and 138: Solutions 121 The given group is th
Page 139 and 140: 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
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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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