A Course In Computational Algebraic Number Theory
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A Course In Computational Algebraic Number Theory
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Author : Henri Cohen
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
A Course In Computational Algebraic Number Theory written by Henri Cohen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.
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.
A Course In Computational Algebraic Number Theory
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Author : Henri Cohen
language : en
Publisher: Copernicus
Release Date : 1993
A Course In Computational Algebraic Number Theory written by Henri Cohen and has been published by Copernicus this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Mathematics categories.
Describes 148 algorithms that are fundamental for number-theoretic computations including computations related to algebraic number theory, elliptic curves, primality testing, and factoring. A complete theoretical introduction is given for each subject, reducing prerequisites to a minimum. The detailed description of each algorithm allows immediate.
Course In Computational Algebraic Number Theory
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Author :
language : en
Publisher:
Release Date : 1993
Course In Computational Algebraic Number Theory written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Algebraic number theory categories.
Computational Algebra And Number Theory
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Author : Wieb Bosma
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Computational Algebra And Number Theory written by Wieb Bosma and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
Computers have stretched the limits of what is possible in mathematics. More: they have given rise to new fields of mathematical study; the analysis of new and traditional algorithms, the creation of new paradigms for implementing computational methods, the viewing of old techniques from a concrete algorithmic vantage point, to name but a few. Computational Algebra and Number Theory lies at the lively intersection of computer science and mathematics. It highlights the surprising width and depth of the field through examples drawn from current activity, ranging from category theory, graph theory and combinatorics, to more classical computational areas, such as group theory and number theory. Many of the papers in the book provide a survey of their topic, as well as a description of present research. Throughout the variety of mathematical and computational fields represented, the emphasis is placed on the common principles and the methods employed. Audience: Students, experts, and those performing current research in any of the topics mentioned above.
Algorithmic Algebraic Number Theory
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Author : M. Pohst
language : en
Publisher: Cambridge University Press
Release Date : 1997-09-25
Algorithmic Algebraic Number Theory written by M. Pohst and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-09-25 with Mathematics categories.
Now in paperback, this classic book is addresssed to all lovers of number theory. On the one hand, it gives a comprehensive introduction to constructive algebraic number theory, and is therefore especially suited as a textbook for a course on that subject. On the other hand many parts go beyond an introduction an make the user familliar with recent research in the field. For experimental number theoreticians new methods are developed and new results are obtained which are of great importance for them. Both computer scientists interested in higher arithmetic and those teaching algebraic number theory will find the book of value.
Computational Algebraic Number Theory
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Author : M.E. Pohst
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Computational Algebraic Number Theory written by M.E. Pohst and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Computational algebraic number theory has been attracting broad interest in the last few years due to its potential applications in coding theory and cryptography. For this reason, the Deutsche Mathematiker Vereinigung initiated an introductory graduate seminar on this topic in Düsseldorf. The lectures given there by the author served as the basis for this book which allows fast access to the state of the art in this area. Special emphasis has been placed on practical algorithms - all developed in the last five years - for the computation of integral bases, the unit group and the class group of arbitrary algebraic number fields. Contents: Introduction • Topics from finite fields • Arithmetic and polynomials • Factorization of polynomials • Topics from the geometry of numbers • Hermite normal form • Lattices • Reduction • Enumeration of lattice points • Algebraic number fields • Introduction • Basic Arithmetic • Computation of an integral basis • Integral closure • Round-Two-Method • Round-Four-Method • Computation of the unit group • Dirichlet's unit theorem and a regulator bound • Two methods for computing r independent units • Fundamental unit computation • Computation of the class group • Ideals and class number • A method for computing the class group • Appendix • The number field sieve • KANT • References • Index
Algebraic Number Theory
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Author : Frazer Jarvis
language : en
Publisher: Springer
Release Date : 2014-06-23
Algebraic Number Theory written by Frazer Jarvis and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-06-23 with Mathematics categories.
This undergraduate textbook provides an approachable and thorough introduction to the topic of algebraic number theory, taking the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the first time that the number field sieve has been considered in a textbook at this level.
Algebraic Number Theory And Fermat S Last Theorem
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Author : Ian Stewart
language : en
Publisher: CRC Press
Release Date : 2025-02-07
Algebraic Number Theory And Fermat S Last Theorem written by Ian Stewart and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-02-07 with Mathematics categories.
Updated to reflect current research and extended to cover more advanced topics as well as the basics, Algebraic Number Theory and Fermat’s Last Theorem, Fifth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers, initially from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work. New to the Fifth Edition Pell's Equation x^2-dy^2=1: all solutions can be obtained from a single `fundamental' solution, which can be found using continued fractions. Galois theory of number field extensions, relating the field structure to that of the group of automorphisms. More material on cyclotomic fields, and some results on cubic fields. Advanced properties of prime ideals, including the valuation of a fractional ideal relative to a prime ideal, localisation at a prime ideal, and discrete valuation rings. Ramification theory, which discusses how a prime ideal factorises when the number field is extended to a larger one. A short proof of the Quadratic Reciprocity Law based on properties of cyclotomic fields. This Valuations and p-adic numbers. Topology of the p-adic integers. Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.
Computational Algebraic Number Theory
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Author : M.E. Pohst
language : en
Publisher: Springer Science & Business Media
Release Date : 1993-09
Computational Algebraic Number Theory written by M.E. Pohst and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-09 with Gardening categories.
Computational algebraic number theory has been attracting broad interest in the last few years due to its potential applications in coding theory and cryptography. For this reason, the Deutsche Mathematiker-Vereinigung initiated an introductory graduate seminar on this topic in Dusseldorf. The lectures given there by the author served as the basis for this book which allows fast access to the state of the art in this area. Special emphasis has been placed on practical algorithms - all developed in the last five years - for the computation of integral bases, the unit group and the class group of arbitrary algebraic number fields. The workshops organized by the Gesselschaft fur mathematische Forschung in cooperation with the Deutsche Mathematiker-Vereinigung (German Mathematics Society) are intended to help, in particular, students and younger mathematicians, to obtain an introduction to fields of current research. Through the means of these well-organized seminars, scientists from other fields can also be introduced to new mathematical ideas. The publication of these workshops in the series DMV SEMINAR will make the material available to an even larger audience.
A Textbook Of Algebraic Number Theory
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Author : Sudesh Kaur Khanduja
language : en
Publisher: Springer Nature
Release Date : 2022-04-26
A Textbook Of Algebraic Number Theory written by Sudesh Kaur Khanduja and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-26 with Mathematics categories.
This self-contained and comprehensive textbook of algebraic number theory is useful for advanced undergraduate and graduate students of mathematics. The book discusses proofs of almost all basic significant theorems of algebraic number theory including Dedekind’s theorem on splitting of primes, Dirichlet’s unit theorem, Minkowski’s convex body theorem, Dedekind’s discriminant theorem, Hermite’s theorem on discriminant, Dirichlet’s class number formula, and Dirichlet’s theorem on primes in arithmetic progressions. A few research problems arising out of these results are mentioned together with the progress made in the direction of each problem. Following the classical approach of Dedekind’s theory of ideals, the book aims at arousing the reader’s interest in the current research being held in the subject area. It not only proves basic results but pairs them with recent developments, making the book relevant and thought-provoking. Historical notes are given at various places. Featured with numerous related exercises and examples, this book is of significant value to students and researchers associated with the field. The book also is suitable for independent study. The only prerequisite is basic knowledge of abstract algebra and elementary number theory.