The Problem of Catalan - Maurice Mignotte

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Résumé :
In 1842 the Belgian mathematician Eug?ne Charles Catalan asked whether 8 and 9 are the only consecutive pure powers of non-zero integers. 160 years after, the question was answered affirmatively by the Swiss mathematician of Romanian origin Preda Mih?ilescu. In other words, 3² ? 2³ = 1 is the only solution of the equation x^p ? y^q = 1 in integers x, y, p, q with xy ? 0 and p, q ? 2. In this book we give a complete and (almost) self-contained exposition of Mih?ilesc?s work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume a very modest background:a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.

Sommaire:
? An Historical Account.- Even Exponents.- Cassels' Relations.- Cyclotomic Fields.- Dirichlet L-Series and Class Number Formulas.- Higher Divisibility Theorems.- Gauss Sums and Stickelberger's Theorem.- Mihailescu's Ideal.- The Real Part of Mihailescu's Ideal.- Cyclotomic units.- Selmer Group and Proof of Catalan's Conjecture.- The Theorem of Thaine.- Baker's Method and Tijdeman's Argument.- Appendix A: Number Fields.- Appendix B: Heights.- Appendix C: Commutative Rings, Modules, Semi-Simplicity.- Appendix D: Group Rings and Characters.- Appendix E: Reduction and Torsion of Finite G-Modules.- Appendix F: Radical Extensions.