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

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112 Chapter 6. Algebraic insolubility of the quintic 2. On the other hand, if a rational function v = v (x1, . . . , xn) satisfies an equation of degree m with symmetric functions of x1, . . . , xn as its coefficients, the function v must have at most m different values under permutations of x1, . . . , xn. If the equation is furthermore known to be irreducible, v must take on exactly m values. Thus, to the relation v = m√ R corresponds an equation of degree m with symmetric coefficients. 6.6 ABEL’s combination of results into an impossibility proof The fourth component of ABEL’S impossibility proof concerned detailed and highly explicit, “computational” investigations of functions of five quantities having two or five values. ABEL sought to reduce all such functions to a few standard forms, an approach completely in line with the classification which opened his paper. These investigations have been subjected to quite a lot of criticism, rethinking, and eventu- ally incorporation into a broader theory, all of which will be dealt with in subsequent chapters. 6.6.1 Careful studies of functions of five quantities The CAUCHY-RUFFINI theorem described in sections 5.6 and 6.4 had ruled out the ex- istence of functions of five quantities which had three or four different values when their arguments were permuted. The remaining relevant (non-symmetric) cases were concerned with functions having two or five values. In the case of two-valued functions, ABEL reduced all such functions to the alternating one which CAUCHY had also studied; and when the function had five values, ABEL could write it as a fourth degree polynomial in one of the variables with coefficients symmetric in the remaining four. Two-valued functions. In order to describe functions of five quantities having two values under permutations, ABEL let v denote such a function of x1, . . . , x5 having the two values v1, v2. Furthermore, he let v ′ denote a second such function (with the values v ′ 1 and v′ 2 ) and formed two further functions t1 = v1 + v2, and t2 = v1v ′ 1 + v2v ′ 2 .
6.6. Combination into an impossibility proof 113 ABEL claimed that the functions t1 and t2 were both symmetric. 33 The two functions v and v ′ were related through these symmetric functions by Then, ABEL chose for v ′ 1 and concluded that v ′ 2 = −v′ 1 v1 = t1v ′ 2 − t2 v ′ 2 − v′ 1 the alternating function s v ′ 1 = s = ∏ 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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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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60 Chapter 5. Towards unsolvable eq
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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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