A Course in Computational Algebraic Number Theory

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Springer Science & Business Media, Apr 17, 2013 - Mathematics - 536 pages
With the advent of powerful computing tools and numerous advances in math ematics, computer science and cryptography, algorithmic number theory has become an important subject in its own right. Both external and internal pressures gave a powerful impetus to the development of more powerful al gorithms. These in turn led to a large number of spectacular breakthroughs. To mention but a few, the LLL algorithm which has a wide range of appli cations, including real world applications to integer programming, primality testing and factoring algorithms, sub-exponential class group and regulator algorithms, etc ... Several books exist which treat parts of this subject. (It is essentially impossible for an author to keep up with the rapid pace of progress in all areas of this subject.) Each book emphasizes a different area, corresponding to the author's tastes and interests. The most famous, but unfortunately the oldest, is Knuth's Art of Computer Programming, especially Chapter 4. The present book has two goals. First, to give a reasonably comprehensive introductory course in computational number theory. In particular, although we study some subjects in great detail, others are only mentioned, but with suitable pointers to the literature. Hence, we hope that this book can serve as a first course on the subject. A natural sequel would be to study more specialized subjects in the existing literature.

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Fundamental NumberTheoretic Algorithms
Algorithms for Linear Algebra and Lattices
Algorithms on Polynomials
Algorithms for Algebraic Number Theory I
Number Theory
Algorithms for Quadratic Fields
Algorithms for Algebraic Number Theory II
Introduction to Elliptic Curves
Factoring in the Dark Ages
Modern Primality Tests
Modern Factoring Methods
Appendix A Packages for Number Theory
Table of Class Numbers and Units of Totally Real Cubic Fields

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