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

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92 Chapter 5. Towards unsolvable equations for the substitution which transformed the permutation A1 into A2 in the above- mentioned way. 91 He then defined ( A1) to be the product of two substitutions (A2 A6 A3 ) and ( A4 A5 ) if it gave the same result as the two applied sequentially,92 in which case CAUCHY wrote � A1 A6 � = � �� A2 A4 Furthermore, he defined the identical substitution and powers of a substitution to have the meanings we still attribute to these concepts today. 93 The smallest integer n such that the n th power of a substitution was the identity substitution, CAUCHY called the degree of the substitution. 94 All these notational advances played a central part in formalizing the manipulations on permutations and were soon generally adopted. LAGRANGE’S Theorem. In order to demonstrate LAGRANGE’S theorem, CAUCHY let K denote an arbitrary expression in n quantities, A3 A5 K = K (x1, . . . , xn) . With N = n!, he labelled the N different permutations of these n quantities A1, . . . , AN. The values which K would acquire when the corresponding substitutions of the form ( A1 Au ) were applied were correspondingly labelled K1, . . . , KN, � � A1 Ku = K for 1 ≤ u ≤ N. Au If these were all distinct, the expression K would obviously have N different values when its arguments were interchanged. In the contrary case, CAUCHY assumed that for M indices the values of K were equal � . Kα = K β = Kγ = . . . . The core of the proof was CAUCHY’S realization that if the permutation A λ was fixed and the substitution ( Aα A β ) was applied to A λ giving Aµ, i.e. � Aα Aµ = A β � A λ, the corresponding values K λ and Kµ would be identical. Consequently, the different values of K came in bundles of M and CAUCHY had deduced that M had to divide n!. The central concept of degree of equivalence, which RUFFINI had introduced to mean 91 (A.-L. Cauchy, 1815a, 67). 92 (ibid., 73). 93 (ibid., 73, 74) 94 (ibid., 76). ABEL was later to change this term to the now standard order.
5.6. CAUCHY’ theory of permutations and a new proof of RUFFINI’s theorem 93 the number of substitutions which left the given function unaltered, was renamed the indicative divisor (French: “diviseur indicatif”) by CAUCHY and was exactly what he had denoted by M. Terming the number of different values of K under all possible substitutions the index of the function K and denoting it by R, CAUCHY had obtained the formula n! = R × M. (5.13) The RUFFINI-CAUCHY Theorem. After explicitly providing the function of n quantities a1, . . . , an given by � � ∏ ai − aj 1≤i
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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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7.1. Solubility of Abelian equation
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7.2. Elliptic functions 153 Figure
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7.2. Elliptic functions 155 7.2.1 T
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7.3. The concept of irreducibility
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7.3. The concept of irreducibility
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7.4. Enlarging the class of solvabl
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164 Chapter 8. A grand theory in sp
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Part III Interlude: ABEL and the
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192 Chapter 9. The nineteenth-centu
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194 Chapter 10. Toward rigorization
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208 Chapter 11. CAUCHY’s new foun
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Chapter 12 ABEL’s reading of CAUC
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12.1. ABEL’s critical attitude 22
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12.3. Convergence 229 12.3 Converge
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12.3. Convergence 231 and {εm} was
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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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