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AuthorRibenboim, Paulo. author
Title13 Lectures on Fermat's Last Theorem [electronic resource] / by Paulo Ribenboim
ImprintNew York, NY : Springer New York, 1979
Connect tohttp://dx.doi.org/10.1007/978-1-4684-9342-9
Descript XVI, 302 p. online resource

CONTENT

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 Naรฏve 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 of Furtwรคngler -- 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


Mathematics Computer science Number theory Mathematics Number Theory Computer Science general



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