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Problem of Catalan / Yuri F. Bilu, Yann Bugeaud and Maurice Mignotte.

By: Contributor(s): Material type: TextTextPublication details: New York : Springer, 2014.Description: xiv, 245 p. ; 25 cmISBN:
  • 9783319100937
Subject(s): DDC classification:
  • 512.922 23 B599
Contents:
1. A historical account -- 2. Even exponents -- 3. Cassels' relations -- 4. Cyclotomic fields -- 5. Dirichlet L-series and class number formulas -- 6. Higher divisibility theorems -- 7. Gauss sums and Stickelberger's theorem -- 8. Mihăilescu's ideal -- 9. The real part of Mihăilescu's ideal -- 10. Cyclotomic units -- 11. Selmer group and proof of Catalan's conjecture -- 12. The theorem of Thaine -- 13. 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-- References-- Author Index-- Subject Index.
Summary: In this book it is given a complete and (almost) self-contained exposition of Mihailescu’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.
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Includes bibliographical references and indexes.

1. A historical account --
2. Even exponents --
3. Cassels' relations --
4. Cyclotomic fields --
5. Dirichlet L-series and class number formulas --
6. Higher divisibility theorems --
7. Gauss sums and Stickelberger's theorem --
8. Mihăilescu's ideal --
9. The real part of Mihăilescu's ideal --
10. Cyclotomic units --
11. Selmer group and proof of Catalan's conjecture --
12. The theorem of Thaine --
13. 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--
References--
Author Index--
Subject Index.

In this book it is given a complete and (almost) self-contained exposition of Mihailescu’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.

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