13 Lectures on Fermat's Last Theorem von Paulo Ribenboim | ISBN 9780387904320

13 Lectures on Fermat's Last Theorem

von Paulo Ribenboim
Buchcover 13 Lectures on Fermat's Last Theorem | Paulo Ribenboim | EAN 9780387904320 | ISBN 0-387-90432-8 | ISBN 978-0-387-90432-0

13 Lectures on Fermat's Last Theorem

von Paulo Ribenboim

Inhaltsverzeichnis

  • Lecture I The Early History of Fermat’s Last Theorem.
  • 1 The Problem.
  • 2 Early Attempts.
  • 3 Kummer’s Monumental Theorem.
  • 4 Regular Primes.
  • 5 Kummer’s Work on Irregular Prime Exponents.
  • 6 Other Relevant Results.
  • 7 The Golden Medal and the Wolfskehl Prize.
  • Lecture II Recent Results.
  • 1 Stating the Results.
  • 2 Explanations.
  • Lecture III B. K. = Before Kummer.
  • 1 The Pythagorean Equation.
  • 2 The Biquadratic Equation.
  • 3 The Cubic Equation.
  • 4 The Quintic Equation.
  • 5 Fermat’s Equation of Degree Seven.
  • Lecture IV The Naïve Approach.
  • 1 The Relations of Barlow and Abel.
  • 2 Sophie Germain.
  • 3 Congruences.
  • 4 Wendt’s Theorem.
  • 5 Abel’s Conjecture.
  • 6 Fermat’s Equation with Even Exponent.
  • 7 Odds and Ends.
  • Lecture V Kummer’s Monument.
  • 1 A Justification of Kummer’s Method.
  • 2 Basic Facts about the Arithmetic of Cyclotomic Fields.
  • 3 Kummer’s Main Theorem.
  • Lecture VI Regular Primes.
  • 1 The Class Number of Cyclotomic Fields.
  • 2 Bernoulli Numbers and Kummer’s Regularity Criterion.
  • 3 Various Arithmetic Properties of Bernoulli Numbers.
  • 4 The Abundance of Irregular Primes.
  • 5 Computation of Irregular Primes.
  • Lecture VII Kummer Exits.
  • 1 The Periods of the Cyclotomic Equation.
  • 2 The Jacobi Cyclotomic Function.
  • 3 On the Generation of the Class Group of the Cyclotomic Field.
  • 4 Kummer’s Congruences.
  • 5 Kummer’s Theorem for a Class of Irregular Primes.
  • 6 Computations of the Class Number.
  • Lecture VIII After Kummer, a New Light.
  • 1 The Congruences of Mirimanoff.
  • 2 The Theorem of Krasner.
  • 3 The Theorems of Wieferich and Mirimanoff.
  • 4 Fermat’s Theorem and the Mersenne Primes.
  • 5 Summation Criteria.
  • 6 Fermat Quotient Criteria.
  • Lecture IX The Power of Class Field Theory.
  • 1 The Power Residue Symbol.
  • 2 Kummer Extensions.
  • 3 The Main Theorems ofFurtwängler.
  • 4 The Method of Singular Integers.
  • 5 Hasse.
  • 6 The p-Rank of the Class Group of the Cyclotomic Field.
  • 7 Criteria of p-Divisibility of the Class Number.
  • 8 Properly and Improperly Irregular Cyclotomic Fields.
  • Lecture X Fresh Efforts.
  • 1 Fermat’s Last Theorem Is True for Every Prime Exponent Less Than 125000.
  • 2 Euler Numbers and Fermat’s Theorem.
  • 3 The First Case Is True for Infinitely Many Pairwise Relatively Prime Exponents.
  • 4 Connections between Elliptic Curves and Fermat’s Theorem.
  • 5 Iwasawa’s Theory.
  • 6 The Fermat Function Field.
  • 7 Mordell’s Conjecture.
  • 8 The Logicians.
  • Lecture XI Estimates.
  • 1 Elementary (and Not So Elementary) Estimates.
  • 2 Estimates Based on the Criteria Involving Fermat Quotients.
  • 3 Thue, Roth, Siegel and Baker.
  • 4 Applications of the New Methods.
  • Lecture XII Fermat’s Congruence.
  • 1 Fermat’s Theorem over Prime Fields.
  • 2 The Local Fermat’s Theorem.
  • 3 The Problem Modulo a Prime-Power.
  • Lecture XIII Variations and Fugue on a Theme.
  • 1 Variation I (In the Tone of Polynomial Functions).
  • 2 Variation II (In the Tone of Entire Functions).
  • 3 Variation III (In the Theta Tone).
  • 4 Variation IV (In the Tone of Differential Equations).
  • 5 Variation V (Giocoso).
  • 6 Variation VI (In the Negative Tone).
  • 7 Variation VII (In the Ordinal Tone).
  • 8 Variation VIII (In a Nonassociative Tone).
  • 9 Variation IX (In the Matrix Tone).
  • 10 Fugue (In the Quadratic Tone).
  • Epilogue.
  • Index of Names.