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

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52 Chapter 2 From this definition it is clear that in the complex numbers there is nothing of the ‘supernatural’: the complex numbers are nothing but pairs of real numbers. However, a question may arise: is it correct to call such objects numbers? We will answer this question at the end of this section. Another question, which perhaps the reader may put, is the reason way the operations of addition and multiplication of complex numbers are defined exactly in this manner and, in particular, way the operation of multiplication is so strange. We will answer this question in §2.3. First, we clarify the remarkable properties of the set of complex numbers which we had defined. 202. Prove that the complex numbers form a commutative group under addition. Which complex number is the unit element (zero) of this group? In the sequel it will be convenient to denote the complex numbers by a single letter, for example by (or 203. Prove that the operation of multiplication of complex numbers is commutative and associative, i.e., that and It is easy to verify that for every complex number The complex number (1, 0) is therefore the unit element in the set of complex numbers under multiplication. 204. Let be an arbitrary complex number and suppose that Prove that there exists a complex number such that The results of Problems 203 and 204 show that complex numbers form a commutative group under multiplication. 205. Prove that the operations of addition and multiplication of complex numbers possess the distributive property, i.e., that for any complex numbers From the results of Problems 202–205 it follows that the complex numbers with the operations of addition and multiplication defined by (2.2) and (2.3) form a field. This field is the field of complex numbers.
The complex numbers 53 For complex numbers of type where is any real number, formulae (2.2) and (2.3) give Consequently if one associates to every complex number of type the real number then to the operations on numbers of type there correspond the usual operations on real numbers. Therefore we simply identify the complex number with the real number and we say that the field of complex numbers contains the field of real numbers 2 . The complex number (0, 1) is not real (under our definition) and we will denote it by Since the field of complex numbers contains all real numbers and the number it also contains all numbers of the form and where and are any two real numbers and the operations of addition and multiplication are extended to operations on complex numbers. 206. Let be a complex number. Prove that From the result of Problem 206 we obviously obtain that if and only if and As a consequence every complex number can be represented in a unique way in the form where and are two real numbers. If then, following tradition, is called the real part of the complex number, the imaginary part, and the coefficient of the imaginary part. The representation of a complex number in the form is called the algebraic representation of For the complex numbers in algebraic representation formulae (2.2) and (2.3) read: 207. Solve the equation (i.e., find the formula for the difference) 2 In an analogous way, for example, one identifies the rational with the integer
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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
Page 17 and 18: Introduction We begin this book by
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: The complex numbers 51 Consequently
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
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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
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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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