Advanced Topics in Computional Number Theory (Graduate Texts in Mathematics) (Hardcover)

by Henri Cohen (Author) "The easiest way to start studying number fields is to consider them per se, as absolute extensions of Q; this is, for example, what we..." (more)
Key Phrases: principal ideal algorithm, integral ideal coprime, relative defining polynomial, Prove Proposition, Algorithmic Characterization of the Set, Basic Algorithms (more...)

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Product Description
The present book addresses a number of specific topics in computational number theory whereby the author is not attempting to be exhaustive in the choice of subjects. The book is organized as follows. Chapters 1 and 2 contain the theory and algorithms concerning Dedekind domains and relative extensions of number fields, and in particular the generalization to the relative case of the round 2 and related algorithms. Chapters 3, 4, and 5 contain the theory and complete algorithms concerning class field theory over number fields. The highlights are the algorithms for computing the structure of (Z_K/m)^*, of ray class groups, and relative equations for Abelian extensions of number fields using Kummer theory. Chapters 1 to 5 form a homogeneous subject matter which can be used for a 6 months to 1 year graduate course in computational number theory. The subsequent chapters deal with more miscellaneous subjects. Written by an authority with great practical and teaching experience in the field, this book together with the author's earlier book will become the standard and indispensable reference on the subject.

Product Details

• Hardcover: 578 pages
• Publisher: Springer; 1 edition (November 30, 1999)
• Language: English
• ISBN-10: 0387987274
• ISBN-13: 978-0387987279
• Product Dimensions: 9.2 x 6.6 x 1.2 inches
• Shipping Weight: 2.2 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars See all reviews (1 customer review)
• Amazon.com Sales Rank: #1,425,690 in Books (See Bestsellers in Books)

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Most Helpful Customer Reviews

1 of 2 people found the following review helpful:

5.0 out of 5 stars A first class job, just as in the first volume, August 25, 2001

By	Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews (TOP 100 REVIEWER)    (REAL NAME)

The author continues his excellent overview of computational number theory in this book. And, as in the first volume, the writing is first-rate and gives the reader a comprehensive overview of the more advanced algorithms in the subject. Contrary to the first volume, I have not used many of the algorithms in this book and cannot attest to their quality, but the author gives the detailed background on each of them, enhancing their credibility. I read this book mostly to gain more insight into algebraic geometry and its connection with coding theory and cryptography. The following algorithms were ones that I found helpful and interesting but only a few of which I coded myself: (1) The algorithm for generating a random element from an ideal. (2) The compositum of two number fields (3) Valuation of a prime ideal. (4)Ideal factorization. (5)Smith normal form for finite groups. (6) Quotient of groups. (7) Group extensions. (8) Right four-term exact sequences. (9) Image and inverse image of a subgroup. (10)Subgroups with prime index. (11) Solving linear systems in integers. (12) Algorithms involving p-adic logarithms. (13) Computation of the Dedekind eta-function. (14) Unramified Abelian extensions using complex multiplication. (15) Computation of quasi-periods for elliptic functions. (16) Computation of the sigma-function for an elliptic curve.