RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

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182 Chapter 8. A grand theory in spe Figure 8.2: EVARISTE GALOIS (1811–1832) in the present context, 41 which focuses on the differences and similarities between the almost concurrent works of ABEL and GALOIS. 8.5.1 The emergence of a general theory of solubility In a sequence of manuscripts, GALOIS attacked the same two problems as ABEL had suggested in order to describe the extension of algebraic solubility (see section 8.1). ABEL had attempted to solve the first problem — that of finding all solvable equations of a given degree — in his notebook manuscript described in this chapter. ABEL’S second question concerning the determination of whether a given equation was algebra- ically solvable or not was the direct purpose of GALOIS’ theory. GALOIS intended to give characterizations of solubility which could, at least in principle, be used to decide the solubility of any given equation, but the machinery needed for actually determining the solubility of given equations was of lesser interest to him. 42 The important feature of GALOIS’ theory was to associate a structure called a group to any given equation such that the question of solubility of equations could be trans- lated into questions concerning these structures. Although the concept of group only saw its first instances and was not a developed abstract concept in the works of GA- 41 It has been dealt with extensively in the literature, for instance (J. Pierpont, 1898), (Kiernan, 1971), (Hirano, 1984), (Scholz, 1990), or (Martini, 1999). 42 (Kiernan, 1971, 83).
8.5. General resolution of the problem by E. GALOIS 183 LOIS, he was instrumental in bringing about the structural approach to mathematics, which came to dominate much of 20 th century mathematics. 43 GALOIS’ work was, as he himself somewhat laconically remarked, 44 founded in the theory of permutations most of which he had taken over from CAUCHY. GALOIS considered an equation of degree m φ (x) = 0 having the roots x1, . . . , xm, and claimed that the group of the equation G — later called the Galois group — could always be found, which had the following two properties: 1. that every function of the roots x1, . . . , xm which was (numerically) invariant under the substitutions of G was rationally known, and conversely, 2. that every rational function of the roots x1, . . . , xm was invariant under the substitutions of G. GALOIS took over the concept of rationally known from LAGRANGE but changed the notion of invariant to stress numerical invariance instead of LAGRANGE’S formal invariance in order to deal with special (i.e. non-general) equations. However, GALOIS’ proof of the existence of the group of the equation suffered from the unclear character of his concept of invariance. 45 Although the concepts of permutation and substitution underwent some uncom- pleted changes in GALOIS’ manuscripts, he clearly perceived the multiplicative nature of substitutions — understood as transitions from one arrangement (permutation) to another — as well as the multiplicative closure of the GALOIS group. “It is clear in the group of permutations under consideration, the arrangement of letters is not important, but only the substitutions on the letters, by which we move from one permutation to another. Thus, if in similar group one has the substitutions S and T, one is also certain to have the substitution ST.” 46 The second component of GALOIS’ theory addressed the reduction of the group of an equation by the adjunction of quantities to the set of rationally known quantities. By adjoining to the rationally known quantities a single root of an irreducible aux- iliary equation, GALOIS could decompose the group of the equation into a number, p, of subgroups. These had the remarkable property that applying a substitution to 43 These aspects of GALOIS’ work have been studied by, for instance, (Wussing, 1969) and (Kiernan, 1971). 44 (Galois, 1830, 165). 45 (Kiernan, 1971, 80–81). 46 “Comme il s’agit toujours de questions où la disposition primitive des lettres n’influe en rien, dans les groupes que nous considérons, on devra avoir les mêmes substitutions quelle que soit la permutation d’où l’on sera parti. Donc si dans un pareil groupe on a les substitutions S et T, on est sûr d’avoir la substitution ST.” (Galois, 1831c, 47). I have extended the translation found in (Kiernan, 1971, 80).
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RePoSS: Research Publications on Sc
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The Mathematics of NIELS HENRIK ABE
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The Mathematics of NIELS HENRIK ABE
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Contents Contents i List of Tables
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8.3 Refocusing on the equation . .
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V ABEL’s mathematics and the rise
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List of Figures 2.1 NIELS HENRIK AB
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List of Boxes 1 The algebraic reduc
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Summary The present PhD dissertatio
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Preface to the 2004 edition For thi
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in connection with the Abel centenn
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items published in the same year ar
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the Mittag-Leffler archives in Djur
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Chapter 1 Introduction In the after
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1.2. The mathematical topics involv
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1.3. Themes from early nineteenth-c
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1.4. Reflections on methodology 9 l
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1.4. Reflections on methodology 11
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18 Chapter 2. Biography of NIELS HE
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Chapter 3 Historical background The
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3.2. ABEL’s position in mathemati
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3.3. The state of mathematics 43 tr
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3.4. ABEL’s legacy 45 As can be s
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Chapter 4 The position and role of
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4.1. Outline of ABEL’s results an
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4.2. Mathematical change as a histo
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4.2. Mathematical change as a histo
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58 Chapter 5. Towards unsolvable eq
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98 Chapter 6. Algebraic insolubilit
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100 Chapter 6. Algebraic insolubili
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Page 219: Part III Interlude: ABEL and the
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Page 251 and 252: Chapter 12 ABEL’s reading of CAUC
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12.4. Continuity 233 it followed th
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12.4. Continuity 235 Actually, if t
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12.4. Continuity 237 DIRICHLET intr
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12.5. ABEL’s “exception” 239
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12.6. A curious reaction: Lehrsatz
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12.7. From power series to absolute
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12.7. From power series to absolute
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12.8. Product theorems of infinite
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12.8. Product theorems of infinite
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12.9. ABEL’s proof of the binomia
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12.10. Aspects of ABEL’s binomial
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12.10. Aspects of ABEL’s binomial
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266 Chapter 13. ABEL and OLIVIER on
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Chapter 14 Reception of ABEL’s co
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14.1. Reception of ABEL’s rigoriz
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14.2. Conclusion 281 of basic notio
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Part IV Elliptic functions and the
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286 Chapter 15. Elliptic integrals
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300 Chapter 16. The idea of inverti
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Chapter 17 Steps in the process of
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17.1. Infinite representations 323
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17.2. Elliptic functions as ratios
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17.2. Elliptic functions as ratios
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17.3. Characterization of ABEL’s
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Chapter 18 Tools in ABEL’s resear
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18.1. Transformation theory 333 The
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18.1. Transformation theory 335 Obv
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18.1. Transformation theory 337 Sum
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18.2. Integration in logarithmic te
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18.3. Conclusion 345 Summary. ABEL
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Chapter 19 The Paris memoir N. H. A
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19.1. ABEL’s approach to the Pari
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19.2. The contents of ABEL’s Pari
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19.3. Additional, tentative remarks
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19.3. Additional, tentative remarks
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19.4. The fate of the Paris memoir
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19.6. Conclusion 377 hyperelliptic
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Chapter 20 General approaches to el
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20.2. Other ways of introducing ell
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20.3. Conclusion 383 All these four
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Chapter 21 ABEL’s mathematics and
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21.1. From formulae to concepts 389
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21.1. From formulae to concepts 391
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21.2. Concepts and classes enter ma
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21.3. The role of counter examples
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21.4. Conclusion 401 It is beyond t
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404 Appendix A. ABEL’s correspond
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Appendix B ABEL’s manuscripts Man
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1. Précis d’une théorie des fon
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Bibliography Abel (MS:351:A). “M
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Bibliography 413 in: Journal für d
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Bibliography 415 — (1990). “Geo
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Bibliography 417 — (1830). “Mé
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Bibliography 419 — (1768). “Ins
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Bibliography 421 Grattan-Guinness,
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Bibliography 423 Jahnke, H. N. (198
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Bibliography 425 Legendre, A. M. (1
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Bibliography 427 Rosen, M. (1981).
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Bibliography 429 Toti Rigatelli, L.
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432 Index of names Frobenius, Georg
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Index École Normale, 40, 187 Écol
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Index 437 Lagrange interpolation, 3
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