Integers, Polynomials, and Rings: A Course in Algebra (Undergraduate Texts in Mathematics) (Hardcover)
~ Irving R.
Details
# Hardcover: 284 pages
# Publisher: Springer; 1 edition (January 8, 2004)
# Language: English
# ISBN-10: 0387403973
# ISBN-13: 978-0387403977
Description
Mathematics is often regarded as the study of calculation, but in fact, mathematics is much more. It combines creativity and logic in order to arrive at abstract truths. This book is intended to illustrate how calculation, creativity, and logic can be combined to solve a range of problems in algebra. Originally conceived as a text for a course for future secondary-school mathematics teachers, this book has developed into one that could serve well in an undergraduate course in abstract algebra or a course designed as an introduction to higher mathematics. Not all topics in a traditional algebra course are covered. Rather, the author focuses on integers, polynomials, their ring structure, and fields, with the aim that students master a small number of serious mathematical ideas. The topics studied should be of interest to all mathematics students and are especially appropriate for future teachers.
One nonstandard feature of the book is the small number of theorems for which full proofs are given. Many proofs are left as exercises, and for almost every such exercise a detailed hint or outline of the proof is provided. These exercises form the heart of the text. Unwinding the meaning of the hint or outline can be a significant challenge, and the unwinding process serves as the catalyst for learning.
Ron Irving is the Divisional Dean of Natural Sciences at the University of Washington. Prior to assuming this position, he served as Chair of the Department of Mathematics. He has published research articles in several areas of algebra, including ring theory and the representation theory of Lie groups and Lie algebras. In 2001, he received the University of Washington's Distinguished Teaching Award for the course on which this book is based.
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Elliptic Curves: Number Theory and Cryptography (Discrete Mathematics and Its Applications)
by: Lawrence C. Washington
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# Hardcover: 440 pages
# Publisher: Chapman & Hall/CRC; 1 edition (May 28, 2003)
# Language: English
# ISBN-10: 1584883650
# ISBN-13: 978-1584883654
Description
Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students.Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired.By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.
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A Course in Arithmetic 1996 (Graduate Texts in Mathematics)
by: Jean Pierre Serre
Details
# Hardcover: 132 pages
# Publisher: Springer (October 10, 1996)
# Language: English
# ISBN-10: 0387900403
# ISBN-13: 978-0387900407
Description
Jean-Pierre Serre is Professor at the Collège de France. He has written a number of books, including "Algebraic Groups and Class Fields", "Local Fields", "Complex Semisimple Lie Algebras", "Linear Representations of Finite Groups", Collected Papers (3 volumes), and "Trees" published by Springer-Verlag.
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A Classical Introduction to Modern Number Theory (GTM) - 2nd Edition
by: Kenneth Ireland Michael Rosen
Details
# Publisher: Springer
# Number Of Pages: 389
# Edition: 2nd edition (1990)
# Pages: 389 ordered pages (401 total pages for ebook) and (412 total page for the hard cover)!
Series: GTM (84)
# Sales Rank: 381488
# ISBN / ASIN: 038797329X
# EAN: 9780387973296
# Binding: Hardcover
# Manufacturer: Springer
# Studio: Springer
# Average Rating: 5
# Total Reviews: 6
Description
Bridging the gap between elementary number theory and the systematic study of advanced topics, A Classical Introduction to Modern Number Theory is a well-developed and accessible text that requires only a familiarity with basic abstract algebra. Historical development is stressed throughout, along with wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. An extensive bibliography and many challenging exercises are also included. This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves.
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Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences)
by: Yu.I. Manin Alexei A. Panchishkin
Details
# Hardcover: 514 pages
# Publisher: Springer; 2nd edition (April 5, 2007)
# Language: English
# ISBN-10: 3540203648
# ISBN-13: 978-3540203643
Description
Introduction to Modern Number Theory surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems, the central ideas of modern theories are exposed. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions.
This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories. Moreover, the authors have added a part dedicated to arithmetical cohomology and noncommutative geometry, a report on point counts on varieties with many rational points, the recent polynomial time algorithm for primality testing, and some others subjects.
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Elementary Number Theory, Group Theory and Ramanujan Graphs (London Mathematical Society Student Texts)
by: Giuliana Davidoff, Peter Sarnak, Alain Valette
Details
# Hardcover: 154 pages
# Publisher: Cambridge University Press (January 27, 2003)
# Language: English
# ISBN-10: 0521824265
# ISBN-13: 978-0521824262
Description
This text is a self-contained study of expander graphs, specifically, their explicit construction. Expander graphs are highly connected but sparse, and while being of interest within combinatorics and graph theory, they can also be applied to computer science and engineering. Only a knowledge of elementary algebra, analysis and combinatorics is required because the authors provide the necessary background from graph theory, number theory, group theory and representation theory. Thus the text can be used as a brief introduction to these subjects and their synthesis in modern mathematics.
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Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach (Undergraduate Texts in Mathematics)
by: William Stein
Details
# Hardcover: 166 pages
# Publisher: Springer; 1 edition (December 3, 2008)
# Language: English
# ISBN-10: 0387855246
# ISBN-13: 978-0387855240
Description
This is a textbook about classical elementary number theory and elliptic curves. The first part discusses elementary topics such as primes, factorization, continued fractions, and quadratic forms, in the context of cryptography, computation, and deep open research problems. The second part is about elliptic curves, their applications to algorithmic problems, and their connections with problems in number theory such as Fermat’s Last Theorem, the Congruent Number Problem, and the Conjecture of Birch and Swinnerton-Dyer. The intended audience of this book is an undergraduate with some familiarity with basic abstract algebra, e.g. rings, fields, and finite abelian groups.
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Elementary Number Theory, Cryptography and Codes (Universitext)
by: M. Welleda Baldoni, Ciro Ciliberto, G.M. Piacentini Cattaneo
Details
* Publisher: Springer
* Number Of Pages: 522
* Publication Date: 2008-12-01
* ISBN-10 / ASIN: 3540691995
* ISBN-13 / EAN: 9783540691990
* Binding: Paperback
Description
In this volume one finds basic techniques from algebra and number theory (e.g. congruences, unique factorization domains, finite fields, quadratic residues, primality tests, continued fractions, etc.) which in recent years have proven to be extremely useful for applications to cryptography and coding theory. Both cryptography and codes have crucial applications in our daily lives, and they are described here, while the complexity problems that arise in implementing the related numerical algorithms are also taken into due account. Cryptography has been developed in great detail, both in its classical and more recent aspects. In particular public key cryptography is extensively discussed, the use of algebraic geometry, specifically of elliptic curves over finite fields, is illustrated, and a final chapter is devoted to quantum cryptography, which is the new frontier of the field. Coding theory is not discussed in full; however a chapter, sufficient for a good introduction to the subject, has been devoted to linear codes. Each chapter ends with several complements and with an extensive list of exercises, the solutions to most of which are included in the last chapter.
Though the book contains advanced material, such as cryptography on elliptic curves, Goppa codes using algebraic curves over finite fields, and the recent AKS polynomial primality test, the authors' objective has been to keep the exposition as self-contained and elementary as possible. Therefore the book will be useful to students and researchers, both in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.) seeking a friendly introduction to the important subjects treated here. The book will also be useful for teachers who intend to give courses on these topics.
Written for: Students and researchers in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.); teachers who intend to give courses on these topics
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Elementary Number Theory
by: Gareth A. Jones, Josephine M. Jones
Details
* Publisher: Springer
* Number Of Pages: 200
* Publication Date: 1998-07-31
* ISBN-10 / ASIN: 3540761977
* ISBN-13 / EAN: 9783540761976
* Binding: Paperback
Description
This book gives an undergraduate-level introduction to Number Theory, with the emphasis on fully explained proofs and examples; exercises (with solutions) are integrated into the text. The first few chapters, covering divisibility, prime numbers and modular arithmetic, assume only basic school algebra, and are therefore suitable for first or second year students as an introduction to the methods of pure mathematics. Elementary ideas about groups and rings (summarised in an appendix) are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares; in particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.
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Elementary Number Theory, 6th Edition
by: David M. Burton
Details
* Publisher: McGraw-Hill Science/Engineering/Math
* Number Of Pages: 448
* Publication Date: 2005-09-27
* Sales Rank: 104598
* ISBN / ASIN: 0073051888
* EAN: 9780073051888
* Binding: Hardcover
* Manufacturer: McGraw-Hill Science/Engineering/Math
* Studio: McGraw-Hill Science/Engineering/Math
* Average Rating: 4
* Total Reviews: 8
Description
Elementary Number Theory, Sixth Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton’s engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.
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