Algebraic Number Theory
Crc Press Inc (Verlag)
978-0-8493-3989-9 (ISBN)
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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.
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)
| Erscheint lt. Verlag | 16.3.1999 |
|---|---|
| Reihe/Serie | Discrete Mathematics and Its Applications |
| Zusatzinfo | 774 equations; 6 Tables, black and white |
| Verlagsort | Bosa Roca |
| Sprache | englisch |
| Maße | 156 x 234 mm |
| Gewicht | 862 g |
| Einbandart | gebunden |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| ISBN-10 | 0-8493-3989-8 / 0849339898 |
| ISBN-13 | 978-0-8493-3989-9 / 9780849339899 |
| Zustand | Neuware |
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