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

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176 Chapter 8. A grand theory in spe If any root existed outside the sequence (8.7), a sequence could be based on this root, and a similar deduction would produce another pair of integers β ′ , k ′ related by β ′ k ′ = µ − 1. However, as ABEL demonstrated, from two such sequences a third one corresponding to β ′′ = gcd (β, β ′ ) could also be constructed with the same property β ′′ k ′′ = µ − 1. ABEL knew that if β = β ′ , the two initial sequences were not distinct. If the two initial sequences were assumed to be maximal, a contraction was obtained, since the sequence corresponding to β ′′ was longer than both the initial sequences. Thus, ABEL had demonstrated that the assumption of a root existing outside the maximal sequence (8.7) led to a contradiction, and therefore all the roots were located in a single chain. Using the same notation as in the Mémoire sur une classe particulière, ABEL wrote the set of roots of P = 0 as s, θ (s) , θ 2 (s) , . . . , θ ν−1 (s) , where θ ν (s) = s, and the equation P = 0 was seen to be a specimen of the class of equations which have become known as Abelian equations (see chapter 7). The first result of ABEL’S research had been to reduce the search for algebraic expressions satisfying an arbitrary equation to the search for expressions satisfying an irreducible one. As SYLOW remarks, 26 the present investigation had led to the further restriction to studying only the possible solutions to irreducible Abelian equations whose degree divided µ − 1. The desired complete characterization of expressions solving irreducible Abelian equations was, however, not undertaken in the notebook study. 8.4 Further ideas on the theory of equations Besides the described reduction to Abelian equations, the notebook manuscript and ABEL’S letters contain other interesting results one of which addressed the form of roots of solvable equations. This result can be seen as an elaboration and rigorization of one of L. EULER’S (1707–1783) claims. The form of roots of solvable equations: rigorizing EULER. At the end of the investigation of possible solutions in the notebook, ABEL found that if an equation was solvable by radicals, its solution would be based on the relationship 26 (L. Sylow, 1902, 21). 1 µ sk = A ν−1 i ∏ u=0 m a kuα µ u for 0 ≤ k ≤ ν − 1,
8.4. Further ideas on the theory of equations 177 Figure 8.1: One of the last pages from ABEL’S notebook manuscript on algebraic solubility (Abel, MS:696, 66). Reproduced from (N. H. Abel, 1902e, facsimile III)
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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 156 and 157: 126 Chapter 6. Algebraic insolubili
Page 171 and 172: Chapter 7 Particular classes of equ
Page 173 and 174: 7.1. Solubility of Abelian equation
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Page 187 and 188: 7.3. The concept of irreducibility
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Page 191: 7.4. Enlarging the class of solvabl
Page 194 and 195: 164 Chapter 8. A grand theory in sp
Page 219: Part III Interlude: ABEL and the
Page 222 and 223: 192 Chapter 9. The nineteenth-centu
Page 224 and 225: 194 Chapter 10. Toward rigorization
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Page 251 and 252: Chapter 12 ABEL’s reading of CAUC
Page 253 and 254: 12.1. ABEL’s critical attitude 22
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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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