代数学の数論とフェルマーの最終定理（テキスト・第５版）<br>Algebraic Number Theory and Fermat's Last Theorem （5TH）

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代数学の数論とフェルマーの最終定理（テキスト・第５版）
Algebraic Number Theory and Fermat's Last Theorem （5TH）

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  • 製本 Hardcover:ハードカバー版／ページ数 486 p.
  • 言語 ENG
  • 商品コード 9781032602257
  • DDC分類 512.74

Full Description

Updated to reflect current research and extended to cover more advanced topics as well as the basics, Algebraic Number Theory and Fermat's Last Theorem, Fifth 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, initially 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 Fifth Edition

Pell's Equation x^2-dy^2=1: all solutions can be obtained from a single `fundamental' solution, which can be found using continued fractions.
Galois theory of number field extensions, relating the field structure to that of the group of automorphisms.
More material on cyclotomic fields, and some results on cubic fields.
Advanced properties of prime ideals, including the valuation of a fractional ideal relative to a prime ideal, localisation at a prime ideal, and discrete valuation rings.
Ramification theory, which discusses how a prime ideal factorises when the number field is extended to a larger one.
A short proof of the Quadratic Reciprocity Law based on properties of cyclotomic fields. This
Valuations and p-adic numbers. Topology of the p-adic integers.

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.

Contents

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.