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13 Lectures on Fermat's Last Theorem
von Paulo RibenboimInhaltsverzeichnis
- 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.