Galois Fields And Galois Rings Made Easy
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| Author | : Maurice Kibler |
| Publisher | : Elsevier |
| Total Pages | : 272 |
| Release | : 2017-09-22 |
| Genre | : Mathematics |
| ISBN | : 0081023510 |
This book constitutes an elementary introduction to rings and fields, in particular Galois rings and Galois fields, with regard to their application to the theory of quantum information, a field at the crossroads of quantum physics, discrete mathematics and informatics.The existing literature on rings and fields is primarily mathematical. There are a great number of excellent books on the theory of rings and fields written by and for mathematicians, but these can be difficult for physicists and chemists to access.This book offers an introduction to rings and fields with numerous examples. It contains an application to the construction of mutually unbiased bases of pivotal importance in quantum information. It is intended for graduate and undergraduate students and researchers in physics, mathematical physics and quantum chemistry (especially in the domains of advanced quantum mechanics, quantum optics, quantum information theory, classical and quantum computing, and computer engineering).Although the book is not written for mathematicians, given the large number of examples discussed, it may also be of interest to undergraduate students in mathematics. - Contains numerous examples that accompany the text - Includes an important chapter on mutually unbiased bases - Helps physicists and theoretical chemists understand this area of mathematics
| Author | : Frank H. P. Fitzek |
| Publisher | : Academic Press |
| Total Pages | : 524 |
| Release | : 2020-05-20 |
| Genre | : Technology & Engineering |
| ISBN | : 0128209046 |
Computing in Communication Networks: From Theory to Practice provides comprehensive details and practical implementation tactics on the novel concepts and enabling technologies at the core of the paradigm shift from store and forward (dumb) to compute and forward (intelligent) in future communication networks and systems. The book explains how to create virtualized large scale testbeds using well-established open source software, such as Mininet and Docker. It shows how and where to place disruptive techniques, such as machine learning, compressed sensing, or network coding in a newly built testbed. In addition, it presents a comprehensive overview of current standardization activities. Specific chapters explore upcoming communication networks that support verticals in transportation, industry, construction, agriculture, health care and energy grids, underlying concepts, such as network slicing and mobile edge cloud, enabling technologies, such as SDN/NFV/ ICN, disruptive innovations, such as network coding, compressed sensing and machine learning, how to build a virtualized network infrastructure testbed on one's own computer, and more. - Provides a uniquely comprehensive overview on the individual building blocks that comprise the concept of computing in future networks - Gives practical hands-on activities to bridge theory and implementation - Includes software and examples that are not only employed throughout the book, but also hosted on a dedicated website
| Author | : Patrick Morandi |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 294 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461240409 |
In the fall of 1990, I taught Math 581 at New Mexico State University for the first time. This course on field theory is the first semester of the year-long graduate algebra course here at NMSU. In the back of my mind, I thought it would be nice someday to write a book on field theory, one of my favorite mathematical subjects, and I wrote a crude form of lecture notes that semester. Those notes sat undisturbed for three years until late in 1993 when I finally made the decision to turn the notes into a book. The notes were greatly expanded and rewritten, and they were in a form sufficient to be used as the text for Math 581 when I taught it again in the fall of 1994. Part of my desire to write a textbook was due to the nonstandard format of our graduate algebra sequence. The first semester of our sequence is field theory. Our graduate students generally pick up group and ring theory in a senior-level course prior to taking field theory. Since we start with field theory, we would have to jump into the middle of most graduate algebra textbooks. This can make reading the text difficult by not knowing what the author did before the field theory chapters. Therefore, a book devoted to field theory is desirable for us as a text. While there are a number of field theory books around, most of these were less complete than I wanted.
| Author | : Andreas Weber |
| Publisher | : MDPI |
| Total Pages | : 160 |
| Release | : 2019-11-04 |
| Genre | : Computers |
| ISBN | : 3039217305 |
Although scientific computing is very often associated with numeric computations, the use of computer algebra methods in scientific computing has obtained considerable attention in the last two decades. Computer algebra methods are especially suitable for parametric analysis of the key properties of systems arising in scientific computing. The expression-based computational answers generally provided by these methods are very appealing as they directly relate properties to parameters and speed up testing and tuning of mathematical models through all their possible behaviors. This book contains 8 original research articles dealing with a broad range of topics, ranging from algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers over methods for certifying the isolated zeros of polynomial systems to computer algebra problems in quantum computing.
| Author | : Juliusz Brzeziński |
| Publisher | : Springer |
| Total Pages | : 296 |
| Release | : 2018-03-21 |
| Genre | : Mathematics |
| ISBN | : 331972326X |
This textbook offers a unique introduction to classical Galois theory through many concrete examples and exercises of varying difficulty (including computer-assisted exercises). In addition to covering standard material, the book explores topics related to classical problems such as Galois’ theorem on solvable groups of polynomial equations of prime degrees, Nagell's proof of non-solvability by radicals of quintic equations, Tschirnhausen's transformations, lunes of Hippocrates, and Galois' resolvents. Topics related to open conjectures are also discussed, including exercises related to the inverse Galois problem and cyclotomic fields. The author presents proofs of theorems, historical comments and useful references alongside the exercises, providing readers with a well-rounded introduction to the subject and a gateway to further reading. A valuable reference and a rich source of exercises with sample solutions, this book will be useful to both students and lecturers. Its original concept makes it particularly suitable for self-study.
| Author | : Jean-Pierre Serre |
| Publisher | : CRC Press |
| Total Pages | : 120 |
| Release | : 2016-04-19 |
| Genre | : Mathematics |
| ISBN | : 1439865256 |
This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. In the first part of the book, classical methods and results, such as the Scholz and Reichardt constructi
| Author | : Francis Borceux |
| Publisher | : Cambridge University Press |
| Total Pages | : 360 |
| Release | : 2001-02-22 |
| Genre | : Mathematics |
| ISBN | : 9780521803090 |
Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context, presenting work by Grothendieck in terms of separable algebras and then proceeding to the infinite-dimensional case, which requires considering topological Galois groups. In the core of the book, the authors first formalize the categorical context in which a general Galois theorem holds, and then give applications to Galois theory for commutative rings, central extensions of groups, the topological theory of covering maps and a Galois theorem for toposes. The book is designed to be accessible to a wide audience: the prerequisites are first courses in algebra and general topology, together with some familiarity with the categorical notions of limit and adjoint functors. The first chapters are accessible to advanced undergraduates, with later ones at a graduate level. For all algebraists and category theorists this book will be a rewarding read.
| Author | : Jean-Pierre Serre |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 215 |
| Release | : 2013-12-01 |
| Genre | : Mathematics |
| ISBN | : 3642591418 |
This is an updated English translation of Cohomologie Galoisienne, published more than thirty years ago as one of the very first versions of Lecture Notes in Mathematics. It includes a reproduction of an influential paper by R. Steinberg, together with some new material and an expanded bibliography.
| Author | : Irving Kaplansky |
| Publisher | : University of Chicago Press |
| Total Pages | : 217 |
| Release | : 1972 |
| Genre | : Mathematics |
| ISBN | : 0226424510 |
This book combines in one volume Irving Kaplansky's lecture notes on the theory of fields, ring theory, and homological dimensions of rings and modules. "In all three parts of this book the author lives up to his reputation as a first-rate mathematical stylist. Throughout the work the clarity and precision of the presentation is not only a source of constant pleasure but will enable the neophyte to master the material here presented with dispatch and ease."—A. Rosenberg, Mathematical Reviews
| Author | : Minjia Shi |
| Publisher | : Academic Press |
| Total Pages | : 320 |
| Release | : 2017-06-12 |
| Genre | : Mathematics |
| ISBN | : 0128133910 |
Codes and Rings: Theory and Practice is a systematic review of literature that focuses on codes over rings and rings acting on codes. Since the breakthrough works on quaternary codes in the 1990s, two decades of research have moved the field far beyond its original periphery. This book fills this gap by consolidating results scattered in the literature, addressing classical as well as applied aspects of rings and coding theory. New research covered by the book encompasses skew cyclic codes, decomposition theory of quasi-cyclic codes and related codes and duality over Frobenius rings. Primarily suitable for ring theorists at PhD level engaged in application research and coding theorists interested in algebraic foundations, the work is also valuable to computational scientists and working cryptologists in the area. - Consolidates 20+ years of research in one volume, helping researchers save time in the evaluation of disparate literature - Discusses duality formulas in the context of Frobenius rings - Reviews decomposition of quasi-cyclic codes under ring action - Evaluates the ideal and modular structure of skew-cyclic codes - Supports applications in data compression, distributed storage, network coding, cryptography and across error-correction
