Algebraic Number Theory

Richard A. Mollin

Algebraic Number Theory
  • 出版商: CRC
  • 出版日期: 1999-03-16
  • 售價: $1,880
  • 貴賓價: 9.8 折 $1,842
  • 語言: 英文
  • 頁數: 504
  • 裝訂: Hardcover
  • ISBN: 0849339898
  • ISBN-13: 9780849339899

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  • Engages readers by offering an historical perspective through the lives of mathematicians who played pivotal roles in developing algebraic number theory
  • Explores in detail the direct, practical application of algebraic number theory to cryptography
  • Provides a rich source of exercises on varying levels designed to enhance, test, and challenge the reader's understandingSolutions manual available with qualifying course adoptions

    From its history as an elegant but abstract area of mathematics, algebraic number theory now takes its place as a useful and accessible study with important real-world practicality. Unique among algebraic number theory texts, this important work offers a wealth of applications to cryptography, including factoring, primality-testing, and public-key cryptosystems.

    A follow-up to Dr. Mollin's popular Fundamental Number Theory with Applications, Algebraic Number Theory provides a global approach to the subject that selectively avoids local theory. Instead, it carefully leads the student through each topic from the level of the algebraic integer, to the arithmetic of number fields, to ideal theory, and closes with reciprocity laws. In each chapter the author includes a section on a cryptographic application of the ideas presented, effectively demonstrating the pragmatic side of theory.

    In this way Algebraic Number Theory provides a comprehensible yet thorough treatment of the material. Written for upper-level undergraduate and graduate courses in algebraic number theory, this one-of-a-kind text brings the subject matter to life with historical background and real-world practicality. It easily serves as the basis for a range of courses, from bare-bones algebraic number theory, to a course rich with cryptography applications, to a course using the basic theory to prove Fermat's Last Theorem for regular primes. Its offering of over 430 exercises with odd-numbered solutions provided in the back of the book and, even-numbered solutions available a separate manual makes this the ideal text for both students and instructors.


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    Table of Contents

    Algebraic Numbers
    Origins and Foundations
    Algebraic Numbers and Number Fields
    Discriminants, Norms, and Traces
    Algebraic Integers and Integral Bases
    Factorization and Divisibility
    Applications of Unique Factorization
    Applications to Factoring Using Cubic Integers
    Arithmetic of Number Fields
    Quadratic Fields
    Cyclotomic Fields
    Units in Number Rings
    Geometry of Numbers
    Dirichlet's Unit Theorem
    Application: The Number Field Sieve
    Ideal Theory
    Properties of Ideals
    PID's and UFD's
    Norms of Ideals
    Ideal Classes-The Class Group
    Class Numbers of Quadratic Fields
    Cyclotomic Fields and Kummer's Theorem--Bernoulli Numbers and Irregular Primes
    Cryptography in Quadratic Fields
    Ideal Decomposition in Extension Fields
    Inertia, Ramification, and Splitting
    The Different and Discriminant
    Galois Theory and Decomposition
    The Kronecker-Weber Theorem
    An Application--Primality Testing
    Reciprocity Laws
    Cubic Reciprocity
    The Biquadratic Reciprocity Law
    The Stickelberger Relation
    The Eisenstein Reciprocity Law
    Elliptic Curves, Factoring, and Primality
    Appendices
    Groups, Modules, Rings, Fields, and Matrices
    Sequences and Series
    Galois Theory (An Introduction with Exercises)
    The Greek Alphabet
    Latin Phrases
    Solutions to Odd-Numbered Exercises
    Bibliograph
    List of Symbols
    Index (over 1,700 entries)

    商品描述(中文翻譯)

    描述
    這本書提供了數學家在發展代數數論方面扮演關鍵角色的歷史觀點，從而吸引讀者的興趣。同時，它詳細探討了代數數論在密碼學中的直接實際應用。書中還提供了豐富的練習題，旨在增強、測試和挑戰讀者的理解能力。對於符合課程要求的採用者，還提供了解答手冊。

    從作為一個優雅但抽象的數學領域的歷史，代數數論現在成為一門有用且易於理解的研究，具有重要的現實實用性。這本重要的書籍在代數數論教材中獨樹一幟，提供了豐富的密碼學應用，包括因數分解、素性測試和公鑰加密系統。

    作為Mollin博士受歡迎的《基礎數論與應用》的續集，《代數數論》提供了一種全球性的方法，有選擇性地避免了局部理論。相反，它仔細地引導學生從代數整數的層次，到數域的算術，再到理想理論，最後以互惠定律作為結尾。在每一章中，作者都包括了一個關於所呈現的思想的密碼學應用的部分，有效地展示了理論的實用面。

    通過這種方式，《代數數論》提供了一種易於理解但全面的教材。這本書適用於高年級本科生和研究生的代數數論課程，並通過歷史背景和現實實用性將主題活靈活現地呈現出來。它可以作為各種課程的基礎，從基礎的代數數論到富含密碼學應用的課程，再到使用基本理論證明正規素數的費馬大定理的課程。書中提供了超過430個練習題，奇數解答在書的後面，偶數解答在另一本單獨的解答手冊中，這使得它成為學生和教師的理想教材。

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