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

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48 Chapter 2 If K is a field then it is possible, as for the field of the real numbers, to consider the polynomials with coefficients in the field K, or, in other words, the polynomials over K. DEFINITION. An expression like ( being a natural number) where are elements of the field K, and is called a polynomial of degree in one variable over K. If is an element of the field K the expression is itself considered as a polynomial over K, and if it represents a polynomial of degree zero, whereas if the degree of this polynomial is considered to be undefined. The elements are called the coefficients of the polynomial (2.1) and the leading coefficient. Two polynomials in one variable are considered to be equal if and only if the coefficients of the terms of the same degree in both polynomials coincide. Let If in the second member of this equation one replaces with an element of the field K and one carries out the calculations indicated, i.e., the operations of addition and multiplication in the field K, one obtains as a result some element of the field K. One thus writes If where 0 is the zero element of field K, one says that is a root of the equation one also says that is a root of the polynomial The polynomials on any field can be added, subtracted, and multiplied. The sum of two polynomials and is a polynomial in which the coefficient of is equal to the sum (in the field K) of the coefficients of in the polynomials and In the same way one defines the difference of two polynomials. It is evident that the degree of the sum or of the difference of two polynomials is not higher than the maximum of the degree of the given polynomials. To calculate the product of the polynomials and one must multiply every monomial of the polynomial by every monomial of the polynomial according to the rule
The complex numbers 49 where is the product in K, and is the usual sum of integer numbers. Afterwards one must sum all the obtained expressions, collecting the monomial where the variable has the same degree, and replacing the sum by the expression If thus 1 Since and (cf., 197) the degree of the product is equal to i.e., the degree of the product of two polynomials (non-zero) is equal to the sum of the degrees of the given polynomials. Taking into account that the operations of addition and multiplication of the elements of the field K possess the commutative, associative, and distributive properties, it is not difficult to verify that the introduced operations of addition and multiplication of polynomials also possess all these properties. If and is any element of the field K, one obtains The polynomials on an arbitrary field K can be divided by one another with a remainder. Dividing the polynomial by the polynomial with a remainder means finding the polynomials (quotient) and (remainder) such that l The coefficient of in the product is equal to hence here we must impose for and for
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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
Page 13 and 14: of the differential equations. Unfo
Page 15 and 16: 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: The complex numbers 47 195. Prove t
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
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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
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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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