Algebraic Number Theory

Richard A. Mollin

  • 出版商: CRC Press
  • 出版日期: 1999-03-16
  • 售價: $1,880
  • 貴賓價: 9.8$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.


    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
    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
    List of Symbols
    Index (over 1,700 entries)