A Course in Computational Algebraic Number Theory

Henri Cohen (Autor)

Buch | Softcover

XXI, 536 Seiten

2010 | 1. Softcover reprint of the original 1st ed. 1993
Springer Berlin (Verlag)
978-3-642-08142-2 (ISBN)

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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.

1. Fundamental Number-Theoretic Algorithms.- 2. Algorithms for Linear Algebra and Lattices.- 3. Algorithms on Polynomials.- 4. Algorithms for Algebraic Number Theory I.- 5. Algorithms for Quadratic Fields.- 6. Algorithms for Algebraic Number Theory II.- 7. Introduction to Elliptic Curves.- 8. Factoring in the Dark Ages.- 9. Modern Primality Tests.- 10. Modern Factoring Methods.- Appendix A. Packages for Number Theory.- Appendix B. Some Useful Tables.- B.1. Table of Class Numbers of Complex Quadratic Fields.- B.2. Table of Class Numbers and Units of Real Quadratic Fields.- B.3. Table of Class Numbers and Units of Complex Cubic Fields.- B.4. Table of Class Numbers and Units of Totally Real Cubic Fields.- B.5. Table of Elliptic Curves.

From the reviews:

H. Cohen

A Course in Computational Algebraic Number Theory

"With numerous advances in mathematics, computer science, and cryptography, algorithmic number theory has become an important subject. Undoubtedly, this book, written by one of the leading authorities in the field, is one of the most beautiful books available on the market."

-ACTA SCIENTIARUM MATHEMATICARUM

"This book is intended to provide material for a three-semester sequence, introductory, graduate course in computational algebraic number theory. ... The book is excellent. ... The book has 75 sections, making it suitable for a three-semester sequence. There are numerous exercises at all levels ... . The bibliography is quite comprehensive and therefore has intrinsic value in its own right. ... chapters bring the student to the frontiers of the field, covering elliptic curves, modern primality testing and modern factoring methods." (Russell Jay Hendel, The MathematicalAssociation of America, January, 2011)

Erscheint lt. Verlag	8.12.2010
Reihe/Serie	Graduate Texts in Mathematics
Zusatzinfo	XXI, 536 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	856 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	Algebra • algorithm • Algorithm analysis and problem complexity • algorithms • Computer • Computer Algebra • Factorisation • Number Theory • Primality • Symbolic Computation
ISBN-10	3-642-08142-8 / 3642081428
ISBN-13	978-3-642-08142-2 / 9783642081422
Zustand	Neuware