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## Details

Document Type: | Book |
---|---|

All Authors / Contributors: |
Ian Stewart; David Orme Tall |

ISBN: | 9781498738392 1498738397 |

OCLC Number: | 910993713 |

Notes: | "A Chapman & Hall book." |

Description: | xix, 322 pages : illustrations ; 24 cm |

Contents: | Algebraic Methods Algebraic Background Rings and Fields Factorization of Polynomials Field Extensions Symmetric Polynomials Modules Free Abelian Groups Algebraic Numbers Algebraic Numbers Conjugates and Discriminants Algebraic Integers Integral Bases Norms and Traces Rings of Integers Quadratic and Cyclotomic Fields Quadratic Fields Cyclotomic Fields Factorization into Irreducibles Historical Background Trivial Factorizations Factorization into Irreducibles Examples of Non-Unique Factorization into Irreducibles Prime Factorization Euclidean Domains Euclidean Quadratic Fields Consequences of Unique Factorization The Ramanujan-Nagell Theorem Ideals Historical Background Prime Factorization of Ideals The Norm of an Ideal Nonunique Factorization in Cyclotomic Fields Geometric Methods Lattices Lattices The Quotient Torus Minkowski's Theorem Minkowski's Theorem The Two-Squares Theorem The Four-Squares Theorem Geometric Representation of Algebraic Numbers The Space Lst Class-Group and Class-Number The Class-Group An Existence Theorem Finiteness of the Class-Group How to Make an Ideal Principal Unique Factorization of Elements in an Extension Ring Number-Theoretic Applications Computational Methods Factorization of a Rational Prime Minkowski Constants Some Class-Number Calculations Table of Class-Numbers Kummer's Special Case of Fermat's Last Theorem Some History Elementary Considerations Kummer's Lemma Kummer's Theorem Regular Primes The Path to the Final Breakthrough The Wolfskehl Prize Other Directions Modular Functions and Elliptic Curves The Taniyama-Shimura-Weil Conjecture Frey's Elliptic Equation The Amateur Who Became a Model Professional Technical Hitch Flash of Inspiration Elliptic Curves Review of Conics Projective Space Rational Conics and the Pythagorean Equation Elliptic Curves The Tangent/Secant Process Group Structure on an Elliptic Curve Applications to Diophantine Equations Elliptic Functions Trigonometry Meets Diophantus Elliptic Functions Legendre and Weierstrass Modular Functions Wiles's Strategy and Recent Developments The Frey Elliptic Curve The Taniyama-Shimura-Weil Conjecture Sketch Proof of Fermat's Last Theorem Recent Developments Appendices Quadratic Residues Quadratic Equations in Zm The Units of Zm Quadratic Residues Dirichlet's Units Theorem Introduction Logarithmic Space Embedding the Unit Group in Logarithmic Space Dirichlet's Theorem Bibliography Index Exercises appear at the end of each chapter. |

Responsibility: | Ian Stewart, University of Warwick, United Kingdom, David Tall, University of Warwick, United Kingdom. |

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## Reviews

*Editorial reviews*

Publisher Synopsis

"It is the discussion of [Fermat's Last Theorem], I think, that sets this book apart from others - there are a number of other texts that introduce algebraic number theory, but I don't know of any others that combine that material with the kind of detailed exposition of FLT that is found here...To summarize and conclude: this is an interesting and attractive book. It would make an attractive text for an early graduate course on algebraic number theory, as well as a nice source of information for people interested in FLT, and especially its connections with algebraic numbers." -Dr. Mark Hunacek, MAA Reviews, June 2016 Praise for Previous Editions "The book remains, as before, an extremely attractive introduction to algebraic number theory from the ideal-theoretic perspective." -Andrew Bremner, Mathematical Reviews, February 2003 Read more...

*User-contributed reviews*