Ian Stewart, David Tall

Algebraic Number Theory and Fermat's Last Theorem

• Broschiertes Buch

Produktbeschreibung

Updated to reflect current research, Algebraic Number Theory and Fermat's Last Theorem, Fourth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics-the quest for a proof of Fermat's Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers from a relatively concrete point of view. Students will see how Wiles's proof of Fermat's Last Theorem opened many new areas for future work.

New to the Fourth Edition

Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper's proof that Z( 14) is Euclidean

Presents an important new result: Mihailescu's proof of the Catalan conjecture of 1844

Revises and expands one chapter into two, covering classical ideas about modular functions and highlighting the new ideas of Frey, Wiles, and others that led to the long-sought proof of Fermat's Last Theorem

Improves and updates the index, figures, bibliography, further reading list, and historical remarks

Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.

Produktdetails

• Verlag: Chapman and Hall/CRC / Taylor & Francis
• 4. Aufl.
• Seitenzahl: 344
• Erscheinungstermin: 30. September 2020
• Englisch
• Abmessung: 229mm x 152mm x 19mm
• Gewicht: 462g
• ISBN-13: 9780367658717
• ISBN-10: 0367658712
• Artikelnr.: 60012066

Autorenporträt

Ian Stewart is Emeritus Professor of Mathematics at the University of Warwick and a Fellow of the Royal Society. He has six honorary doctorates and is an honorary wizard of Unseen University. His more than 130 books include Professor Stewart's Cabinet of Mathematical Curiosities and the four-volume series The Science of Discworld with Terry Pratchett and Jack Cohen. His SF novels include the trilogy Wheelers, Heaven, and Oracle (with Jack Cohen), The Living Labyrinth and Rock Star (with Tim Poston), and Jack of All Trades. Short story collections are Message from Earth and Pasts, Presents, Futures. His Flatland sequel Flatterland has extensive fantasy elements. He has published 33 short stories in Analog, Omni, Interzone, and Nature, with 10 stories in Nature's 'Futures' series. He was Guest of Honour at Novacon 29 in 1999 and Science Guest of Honour and Hugo Award Presenter at Worldcon 75 in Helsinki in 2017. He delivered the 1997 Christmas Lectures for BBC television. His awards include the Royal Society's Faraday Medal, the Gold Medal of the IMA, the Zeeman Medal, the Lewis Thomas Prize, the Euler Book Prize, the Premio Internazionale Cosmos, the Chancellor's Medal of the University of Warwick, and the Bloody Stupid Johnson Award for Innovative Uses of Mathematics.

Inhaltsangabe

Algebraic Methods: Algebraic Background. Algebraic Numbers. Quadratic and Cylclotomic Fields. Factorization into Irreducibles. Ideals. Geometric Methods: Lattices. Minkowski's Theorem. Geometric Representation of Algebraic Numbers. Class-Group and Class-Number. Number-Theoretic Applications: Computational Methods. Kummer's Special Case of Fermat's Last Theorem. The Path to the Final Breakthrough. Elliptic Curves. Elliptic Functions. Wiles's Strategy and Recent Developments. Appendices: Quadratic Residues. Dirichlet's Units Theorems.

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I. Algebraic Methods. 1. Algebraic Background. 2. Algebraic Numbers. 3.
Quadratic and Cyclotomic Fields. 4. Pell's Equation. 5. Factorisation into
Irreducibles. 6. Ideals. II. Geometric Methods. 7. Lattices. 8. Minkowski's
Theorem. 9. Geometric Representation of Algebraic Numbers. 10. Dirichlet's
Units Theorem. 11. Class-Group and Class-Number. III. Number-Theoretic
Applications. 12. Computational Methods. 13. Kummer's Special Case of
Fermat's Last Theorem. IV. Elliptic Curves and Elliptic Functions. 14.
Elliptic Curves. 15. Elliptic Functions. V. Wiles's Proof of Fermat's Last
Theorem. 16. The Path to the Final Breakthrough. 17. Wiles's Strategy and
Subsequent Developments. VI. Galois Theory and Other Topics. 18. Extensions
and Galois Theory. 19. Cyclotomic and Cubic Fields. 20. Prime Ideals
Revisited. 21. Ramification Theory. 22. Quadratic Reciprocity. 23.
Valuations and p-adic Numbers.

Rezensionen

"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