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Divisor Theory
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Book Synopsis Divisor Theory by : Harold M. Edwards
Download or read book Divisor Theory written by Harold M. Edwards and published by Springer Science & Business Media. This book was released on 2013-06-01 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: Man sollte weniger danach streben, die Grenzen der mathe matischen Wissenschaften zu erweitern, als vielmehr danach, den bereits vorhandenen Stoff aus umfassenderen Gesichts punkten zu betrachten - E. Study Today most mathematicians who know about Kronecker's theory of divisors know about it from having read Hermann Weyl's lectures on algebraic number theory [We], and regard it, as Weyl did, as an alternative to Dedekind's theory of ideals. Weyl's axiomatization of what he calls "Kronecker's" theory is built-as Dedekind's theory was built-around unique factor ization. However, in presenting the theory in this way, Weyl overlooks one of Kronecker's most valuable ideas, namely, the idea that the objective of the theory is to define greatest com mon divisors, not to achieve factorization into primes. The reason Kronecker gave greatest common divisors the primary role is simple: they are independent of the ambient field while factorization into primes is not. The very notion of primality depends on the field under consideration-a prime in one field may factor in a larger field-so if the theory is founded on factorization into primes, extension of the field entails a completely new theory. Greatest common divisors, on the other hand, can be defined in a manner that does not change at all when the field is extended (see {sect}1.16). Only after he has laid the foundation of the theory of divisors does Kronecker consider factorization of divisors into divisors prime in some specified field
Book Synopsis Divisor Theory in Module Categories by :
Download or read book Divisor Theory in Module Categories written by and published by Elsevier. This book was released on 2011-08-26 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: Divisor Theory in Module Categories
Book Synopsis Divisors and Sandpiles by : Scott Corry
Download or read book Divisors and Sandpiles written by Scott Corry and published by American Mathematical Soc.. This book was released on 2018-07-23 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: Divisors and Sandpiles provides an introduction to the combinatorial theory of chip-firing on finite graphs. Part 1 motivates the study of the discrete Laplacian by introducing the dollar game. The resulting theory of divisors on graphs runs in close parallel to the geometric theory of divisors on Riemann surfaces, and Part 1 culminates in a full exposition of the graph-theoretic Riemann-Roch theorem due to M. Baker and S. Norine. The text leverages the reader's understanding of the discrete story to provide a brief overview of the classical theory of Riemann surfaces. Part 2 focuses on sandpiles, which are toy models of physical systems with dynamics controlled by the discrete Laplacian of the underlying graph. The text provides a careful introduction to the sandpile group and the abelian sandpile model, leading ultimately to L. Levine's threshold density theorem for the fixed-energy sandpile Markov chain. In a precise sense, the theory of sandpiles is dual to the theory of divisors, and there are many beautiful connections between the first two parts of the book. Part 3 addresses various topics connecting the theory of chip-firing to other areas of mathematics, including the matrix-tree theorem, harmonic morphisms, parking functions, M-matrices, matroids, the Tutte polynomial, and simplicial homology. The text is suitable for advanced undergraduates and beginning graduate students.
Book Synopsis Divisor Theory in Module Categories by : W. V. Vasconcelos
Download or read book Divisor Theory in Module Categories written by W. V. Vasconcelos and published by Elsevier. This book was released on 2016-06-03 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: North-Holland Mathematics Studies, 14: Divisor Theory in Module Categories focuses on the principles, operations, and approaches involved in divisor theory in module categories, including rings, divisors, modules, and complexes. The book first takes a look at local algebra and homology of local rings. Discussions focus on Gorenstein rings, Euler characteristics of modules, Macaulay rings, Koszul complexes, Noetherian and coherent rings, flatness, and Fitting's invariants. The text then explains divisorial ideals, including divisors, modules of dimension one, and higher divisorial ideals. The manuscript ponders on spherical modules and divisors and I-divisors. Topics include construction, Euler characteristics of Inj (A), change of rings and dimensions, spherical modules, resolutions and divisors, and elementary properties. The text is a valuable source of information for mathematicians and researchers interested in divisor theory in module categories.
Book Synopsis The Divisor Class Group of a Krull Domain by : Robert M. Fossum
Download or read book The Divisor Class Group of a Krull Domain written by Robert M. Fossum and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 157 pages. Available in PDF, EPUB and Kindle. Book excerpt: There are two main purposes for the wntmg of this monograph on factorial rings and the associated theory of the divisor class group of a Krull domain. One is to collect the material which has been published on the subject since Samuel's treatises from the early 1960's. Another is to present some of Claborn's work on Dedekind domains. Since I am not an historian, I tread on thin ice when discussing these matters, but some historical comments are warranted in introducing this material. Krull's work on finite discrete principal orders originating in the early 1930's has had a great influence on ring theory in the suc ceeding decades. Mori, Nagata and others worked on the problems Krull suggested. But it seems to me that the theory becomes most useful after the notion of the divisor class group has been made func torial, and then related to other functorial concepts, for example, the Picard group. Thus, in treating the group of divisors and the divisor class group, I have tried to explain and exploit the functorial properties of these groups. Perhaps the most striking example of the exploitation of this notion is seen in the works of I. Danilov which appeared in 1968 and 1970.
Book Synopsis Algebraic Number Theory by : H. Koch
Download or read book Algebraic Number Theory written by H. Koch and published by Springer Science & Business Media. This book was released on 1997-09-12 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews of the first printing, published as Volume 62 of the Encyclopaedia of Mathematical Sciences: "... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes." Monatshefte fuer Mathematik, 1994 "... Koch's book is written mostly for non-specialists. It is an up-to-date account of the subject dealing with mostly general questions. Special results appear only as illustrating examples for the general features of the theory. It is supposed that the reader has good general background in the fields of modern (abstract) algebra and elementary number theory. We recommend this volume mainly to graduate studens and research mathematicians." Acta Scientiarum Mathematicarum, 1993
Book Synopsis Multiplicative Ideal Theory in Commutative Algebra by : James W. Brewer
Download or read book Multiplicative Ideal Theory in Commutative Algebra written by James W. Brewer and published by Springer Science & Business Media. This book was released on 2006-12-15 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, a tribute to the work of Robert Gilmer, consists of twenty-four articles authored by his most prominent students and followers. These articles combine surveys of past work by Gilmer and others, recent results which have never before seen print, open problems, and extensive bibliographies. The entire collection provides an in-depth overview of the topics of research in a significant and large area of commutative algebra.
Book Synopsis 3264 and All That by : David Eisenbud
Download or read book 3264 and All That written by David Eisenbud and published by Cambridge University Press. This book was released on 2016-04-14 with total page 633 pages. Available in PDF, EPUB and Kindle. Book excerpt: 3264, the mathematical solution to a question concerning geometric figures.
Download or read book Number Theory written by and published by Academic Press. This book was released on 1986-05-05 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used. We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. We start from concrete problems in number theory. General theories arise as tools for solving these problems. As a rule, these theories are developed sufficiently far so that the reader can see for himself their strength and beauty, and so that he learns to apply them. Most of the questions that are examined in this book are connected with the theory of diophantine equations - that is, with the theory of the solutions in integers of equations in several variables. However, we also consider questions of other types; for example, we derive the theorem of Dirichlet on prime numbers in arithmetic progressions and investigate the growth of the number of solutions of congruences.
Download or read book Intersection Theory written by W. Fulton and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 483 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the ancient origins of algebraic geometry in the solution of polynomial equations, through the triumphs of algebraic geometry during the last two cen turies, intersection theory has played a central role. Since its role in founda tional crises has been no less prominent, the lack of a complete modern treatise on intersection theory has been something of an embarrassment. The aim of this book is to develop the foundations of intersection theory, and to indicate the range of classical and modern applications. Although a comprehensive his tory of this vast subject is not attempted, we have tried to point out some of the striking early appearances of the ideas of intersection theory. Recent improvements in our understanding not only yield a stronger and more useful theory than previously available, but also make it possible to devel op the subject from the beginning with fewer prerequisites from algebra and algebraic geometry. It is hoped that the basic text can be read by one equipped with a first course in algebraic geometry, with occasional use of the two appen dices. Some of the examples, and a few of the later sections, require more spe cialized knowledge. The text is designed so that one who understands the con structions and grants the main theorems of the first six chapters can read other chapters separately. Frequent parenthetical references to previous sections are included for such readers. The summaries which begin each chapter should fa cilitate use as a reference.
Book Synopsis Ideal Systems by : Franz Halter-Koch
Download or read book Ideal Systems written by Franz Halter-Koch and published by CRC Press. This book was released on 1998-04-21 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Provides for the first time a concise introduction to general and multiplicative ideal theory, valid for commutative rings and monoids and presented in the language of ideal systems on (commutative) monoids."
Book Synopsis Factorization in Integral Domains by : Daniel Anderson
Download or read book Factorization in Integral Domains written by Daniel Anderson and published by Routledge. This book was released on 2017-11-13 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: The contents in this work are taken from both the University of Iowa's Conference on Factorization in Integral Domains, and the 909th Meeting of the American Mathematical Society's Special Session in Commutative Ring Theory held in Iowa City. The text gathers current work on factorization in integral domains and monoids, and the theory of divisibility, emphasizing possible different lengths of factorization into irreducible elements.
Book Synopsis Representations of Algebras by : Graham J. Leuschke
Download or read book Representations of Algebras written by Graham J. Leuschke and published by American Mathematical Soc.. This book was released on 2018 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held in August 2016, at Syracuse University. This volume includes three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories.
Book Synopsis History Algebraic Geometry by : Jean Dieudonné
Download or read book History Algebraic Geometry written by Jean Dieudonné and published by CRC Press. This book was released on 1985-05-30 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains several fundamental ideas that are revived time after time in different guises, providing a better understanding of algebraic geometric phenomena. It shows how the field is enriched with loans from analysis and topology and from commutative algebra and homological algebra.
Book Synopsis Advances in Ring Theory and Applications by : Shakir Ali
Download or read book Advances in Ring Theory and Applications written by Shakir Ali and published by Springer Nature. This book was released on with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Reviews in Number Theory, 1984-96 by :
Download or read book Reviews in Number Theory, 1984-96 written by and published by . This book was released on 1997 with total page 804 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Combinatorial Algebraic Geometry by : Gregory G. Smith
Download or read book Combinatorial Algebraic Geometry written by Gregory G. Smith and published by Springer. This book was released on 2017-11-17 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consolidates selected articles from the 2016 Apprenticeship Program at the Fields Institute, part of the larger program on Combinatorial Algebraic Geometry that ran from July through December of 2016. Written primarily by junior mathematicians, the articles cover a range of topics in combinatorial algebraic geometry including curves, surfaces, Grassmannians, convexity, abelian varieties, and moduli spaces. This book bridges the gap between graduate courses and cutting-edge research by connecting historical sources, computation, explicit examples, and new results.
