CiNii 図書 - Algebraic number theory

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)

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Algebraic number theory

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