Written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Chapters one through five form a homogenous subject matter suitable for a six-month or year-long course in computational number theory. The subsequent chapters deal with more miscellaneous subjects.
Advanced Topics in Computional Number Theory
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"Das vorliegende Buch ist eine Fortsetzung des bekannten erkes "A Course in Computational Algebraic Number Theory" (Graduate Texts in Mathematics 138) desselben Autors. ... So ist das vorliegende Buch ein sehr umfängliches Nachschlagewerk zur algorithmischen Zahlentheorie, das zusammen mit dem ersten Buch des Autors sicherlich eine Standard-Referenz für zahlentheoretische Algorithmen darstellen wird." Internationale Mathematische Nachrichten, Nr. 187, August 2001
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2 of 3 people found the following review helpful:
5.0 out of 5 stars
A first class job, just as in the first volume,
By Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews (VINE VOICE) (HALL OF FAME REVIEWER) (REAL NAME)
This review is from: Advanced Topics in Computional Number Theory (Graduate Texts in Mathematics) (Hardcover)
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.
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Inside This Book
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First Sentence:
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 have done in [Coh0]. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
principal ideal algorithm, integral ideal coprime, relative defining polynomial, complex cubic fields, relative class groups, squarefree ideals, class field corresponding, discrete logarithm algorithm, relative quadratic extensions, sextic fields, ray class groups, totally real case, cubic number field, class group computations, ray class fields, ideals coprime, quartic fields, small discriminant, prime ideal decomposition, binary cubic forms, real cubic fields, discriminant ideal, gamma product, congruence subgroup, quartic case Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Proposition, Algorithmic Characterization of the Set, Basic Algorithms, Compute Hessian New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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 have done in [Coh0]. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
principal ideal algorithm, integral ideal coprime, relative defining polynomial, complex cubic fields, relative class groups, squarefree ideals, class field corresponding, discrete logarithm algorithm, relative quadratic extensions, sextic fields, ray class groups, totally real case, cubic number field, class group computations, ray class fields, ideals coprime, quartic fields, small discriminant, prime ideal decomposition, binary cubic forms, real cubic fields, discriminant ideal, gamma product, congruence subgroup, quartic case Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Proposition, Algorithmic Characterization of the Set, Basic Algorithms, Compute Hessian New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
See all 100 books this book cites
- Cyclotomic Fields (Graduate Texts in Mathematics) by S. Lang in Front Matter, Back Matter (1), and Back Matter (2)
- Algebraic Number Theory (Graduate Texts in Mathematics) by Serge Lang in Back Matter (1), Back Matter (2), and Back Matter (3)
- Elliptic Curves: Diophantine Analysis (Grundlehren der mathematischen Wissenschaften) by S. Lang in Back Matter (1), and Back Matter (2)
- Number Theory by Helmut Hasse in Back Matter (1), and Back Matter (2)
- An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman in Back Matter (1), and Back Matter (2)
See all 100 books this book cites
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