03432nam 22006255 450 991029998110332120200706113843.03-319-10094-710.1007/978-3-319-10094-4(CKB)3710000000261935(EBL)1965396(OCoLC)893117743(SSID)ssj0001372751(PQKBManifestationID)11785454(PQKBTitleCode)TC0001372751(PQKBWorkID)11310892(PQKB)10600316(MiAaPQ)EBC1965396(DE-He213)978-3-319-10094-4(PPN)18209412X(EXLCZ)99371000000026193520141009d2014 u| 0engur|n|---|||||txtccrThe Problem of Catalan /by Yuri F. Bilu, Yann Bugeaud, Maurice Mignotte1st ed. 2014.Cham :Springer International Publishing :Imprint: Springer,2014.1 online resource (251 p.)Description based upon print version of record.3-319-10093-9 Includes bibliographical references and indexes.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 -- Mihăilescu’s Ideal -- The Real Part of Mihăilescu’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.In 1842 the Belgian mathematician Eugène 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 Mihăilescu. In this book we give a complete and (almost) self-contained exposition of Mihăilescu’s work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume very modest background: a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.Number theoryAlgebraNumber Theoryhttps://scigraph.springernature.com/ontologies/product-market-codes/M25001General Algebraic Systemshttps://scigraph.springernature.com/ontologies/product-market-codes/M1106XAlgebrahttps://scigraph.springernature.com/ontologies/product-market-codes/M11000Number theory.Algebra.Number Theory.General Algebraic Systems.Algebra.511/.6512.72Bilu Yuri Fauthttp://id.loc.gov/vocabulary/relators/aut721239Bugeaud Yannauthttp://id.loc.gov/vocabulary/relators/autMignotte Mauriceauthttp://id.loc.gov/vocabulary/relators/autBOOK9910299981103321The Problem of Catalan2540401UNINA