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

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138 Chapter 6. Algebraic insolubility of the quintic The succeeding part of KRONECKER’S proof concerned the substitution theoretic aspects of ABEL’S proof and consisted of an extended version of the CAUCHY-RUFFINI theorem. For n > 4, KRONECKER let f designate a function of quantities x1, . . . , xn and studied the conjugate functions f1, . . . , fm; these functions were the analogous of what ABEL had called the different values of f under all permutations of x1, . . . , xn. KRO- NECKER derived the result that for any non-symmetric function f , some permutation would exist which altered the value of one of the conjugate functions. He could even demonstrate that if only the n! � � 2 permutations, which left the product ∏i
6.10. Summary 139 of ABEL’S argument but superior in rigour. The classification of algebraic expressions was also a concern of some later 19 th century mathematicians, until it was settled by KÖNIGSBERGER and KRONECKER. The fact that global criticism of ABEL’S impossibility proof was limited can be taken as a sign that the mathematical community soon came to realize the overall validity of the result. The change of attitude toward the problem, which had been facilitated by the statements of experts such as LAGRANGE and GAUSS (section 5.4) and the proofs of RUFFINI, which at least were known in some circles in Paris, had been a prerequisite for the quick acceptance. However, local criticism was still conducted in an effort to make ABEL’S proof clearer and more powerful. Central lemmata, on which doubt could be cast, were reexamined and new proofs were given. 6.10 Summary As described, ABEL’S proof of the insolubility of the general quintic was a curious combination of general theorems and investigations of particular cases. Partly because of the counter intuitive nature of the result and partly because of legitimate local objections to ABEL’S argument, the result was subsequently scrutinized. Inter- preted in terms of delineation of concepts, the algebraic insolubility of the general quintic distinguished the concepts of polynomial equations and algebraically solvable equations.
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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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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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