| Description |
1 online resource |
| Contents |
880-01 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 |
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880-01/(S Machine generated contents note: 1. Historical Account -- 1.1. Catalan's Note Extraite -- 1.2. Particular Cases -- 1.3. Cassels' Relations -- 1.4. Analysis: Logarithmic Forms -- 1.5. Algebra: Cyclotomic Fields -- 1.6. Numerical Results -- 1.7. Final Attack -- 2. Even Exponents -- 2.1. Equation xp = y2 + 1 -- 2.2. Units of Real Quadratic Rings -- 2.3. Equation x2 -- y" = 1 with Q [≥] 5 -- 2.4. Cubic Field Q(3[√]2) -- 2.5. Equation x2 -- y3 = 1 -- 3. Cassels' Relations -- 3.1. Cassels' Divisibility Theorem and Cassels' Relations -- 3.2. Binomial Power Series -- 3.3. Proof of the Divisibility Theorem -- 3.4. Hyyro's Lower Bounds -- 4. Cyclotomic Fields -- 4.1. Degree and Galois Group -- 4.2. Integral Basis and Discriminant -- 4.3. Decomposition of Primes -- 4.4. Units -- 4.5. Real Cyclotomic Field and the Class Group -- 4.6. Cyclotomic Extensions of Number Fields -- 4.7. General Cyclotomic Fields -- 5. Dirichlet L-Series and Class Number Formulas -- 5.1. Dirichlet Characters and L-Series -- 5.2. Dedekind ζ-Function of the Cyclotomic Field -- 5.3. Calculating L(1, χ) for χ [≠] 1 -- 5.4. Class Number Formulas -- 5.5. Composite Moduli -- 6. Higher Divisibility Theorems -- 6.1. Most Important Lemma -- 6.2. Inkeri's Divisibility Theorem -- 6.3. Deviation: Catalan's Problem with Exponent 3 -- 6.4. Group Ring -- 6.5. Stickelberger, Mihailescu, and Wieferich -- 7. Gauss Sums and Stickelberger's Theorem -- 7.1. Stickelberger's Ideal and Stickelberger's Theorem -- 7.2. Gauss Sums -- 7.3. Multiplicative Combinations of Gauss Sums -- 7.4. Prime Decomposition of a Gauss Sum -- 7.5. Proof of Stickelberger's Theorem -- 7.6. Kummer's Basis -- 7.7. Real and the Relative Part of Stickelberger's Ideal -- 7.8. Proof of Iwasawa's Class Number Formula -- 8. Mihailescu's Ideal -- 8.1. Definitions and Main Theorems -- 8.2. Algebraic Number (x -- ζ) -- 8.3. qth Root of (x -- ζ) -- 8.4. Proof of Theorem 8.2 -- 8.5. Proof of Theorem 8.4 -- 8.6. Application to Catalan's Problem I: Divisibility of the Class Number -- 8.7. Application to Catalan's Problem II: Mihailescu's Ideal vs Stickelberger's Ideal -- 8.8. On the Real Part of Mihailescu's Ideal -- 9. Real Part of Mihailescu's Ideal -- 9.1. Main Theorem -- 9.2. Products of Binomial Power Series -- 9.3. Mihailescu's Series (1 + ζ T)/q -- 9.4. Proof of Theorem 9.2 -- 10. Cyclotomic Units -- 10.1. Circulant Determinant -- 10.2. Cyclotomic Units -- 11. Selmer Group and Proof of Catalan's Conjecture -- 11.1. Selmer Group -- 11.2. Selmer Group as Galois Module -- 11.3. Units as Galois Module -- 11.4. q-Primary Cyclotomic Units -- 11.5. Proof of Theorem 11.5 -- 12. Theorem of Thaine -- 12.1. Introduction -- 12.2. Preparations -- 12.3. Proof of Theorem 12.2 -- 12.4. Reduction of a Multiplicative Group Modulo a Prime Ideal -- 12.5. Reduction of a Multiplicative Group Modulo a Prime Number and Proof of Theorem 12.3 -- 13. Baker's Method and Tijdeman's Argument -- 13.1. Introduction: Thue, Gelfond, and Baker -- 13.2. Heights in Finitely Generated Groups -- 13.3. Almost nth Powers -- 13.4. Effective Analysis of Classical Diophantine Equations -- 13.5. Theorem of Schinzel and Tijdeman and the Equation of Pillai -- 13.6. Tijdeman's Argument -- Appendix A Number Fields -- A.1. Embeddings, Integral Bases, and Discriminant -- A.2. Units, Regulator -- A.3. Ideals, Factorization -- A.4. Norm of an Ideal -- A.5. Ideal Classes, the Class Group -- A.6. Prime Ideals, Ramification -- A.7. Galois Extensions -- A.8. Valuations -- A.9. Dedekind ζ-Function -- A.10. Chebotarev Density Theorem -- A.11. Hilbert Class Field -- Appendix B Heights -- Appendix C Commutative Rings, Modules, and Semi-simplicity -- C.1. Cyclic Modules -- C.2. Finitely Generated Modules -- C.3. Semi-simple Modules -- C.4. Semi-simple Rings -- C.5. "Dual" Module -- Appendix D Group Rings and Characters -- D.1. Weight Function and the Norm Element -- D.2. Characters of a Finite Abelian Group -- D.3. Conjugate Characters -- D.4. Semi-simplicity of the Group Ring -- D.5. Idempotents -- Appendix E Reduction and Torsion of Finite G -Modules -- E.1. Telescopic Rings -- E.2. Products of Telescopic Rings -- E.3. Elementary Divisors and Finitely Generated Modules -- E.4. Reduction and Torsion -- Appendix F Radical Extensions -- F.1. Field Generated by a Single Root -- F.2. Kummer's Theory -- F.3. General Radical Extensions -- F.4. Equivariant Kummer's Theory |
| Summary |
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 |
| Analysis |
wiskunde |
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mathematics |
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getallenleer |
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number theory |
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algebra |
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Mathematics (General) |
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Wiskunde (algemeen) |
| Notes |
Online resource; title from PDF title page (EBSCO, viewed October 16, 2014) |
| Subject |
Consecutive powers (Algebra)
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Problem solving.
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Problem Solving
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MATHEMATICS -- Algebra -- Intermediate.
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Consecutive powers (Algebra)
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Problem solving
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| Form |
Electronic book
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| Author |
Bugeaud, Yann, 1971- author.
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Mignotte, Maurice, author.
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| ISBN |
9783319100944 |
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3319100947 |
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3319100939 |
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9783319100937 |
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