*Like its popular predecessors, this text develops ring theory first by drawing on students' familiarity with integers and polynomials.*

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# A First Course in Abstract Algebra

Like its popular predecessors, A First Course in Abstract Algebra: Rings, Groups, and Fields, Third Edition develops ring theory first by drawing on students' familiarity with integers and polynomials. This unique approach motivates students in the study of abstract algebra and helps them understand the power of abstraction. The authors introduce g
# A First Course in Abstract Algebra

Considered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures.
# A First Course in Abstract Algebra

This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each. Number Theory: Induction; Binomial Coefficients; Greatest Common Divisors; The Fundamental Theorem of Arithmetic Congruences; Dates and Days. Groups I: Some Set Theory; Permutations; Groups; Subgroups and Lagrange's Theorem; Homomorphisms; Quotient Groups; Group Actions; Counting with Groups.Commutative Rings I: First Properties; Fields; Polynomials; Homomorphisms; Greatest Common Divisors; Unique Factorization; Irreducibility; Quotient Rings and Finite Fields; Officers, Magic, Fertilizer, and Horizons.Linear Algebra: Vector Spaces; Euclidean Constructions; Linear Transformations; Determinants; Codes; Canonical Forms.Fields: Classical Formulas; Insolvability of the General Quintic; Epilog. Groups II: Finite Abelian Groups; The Sylow Theorems; Ornamental Symmetry. Commutative Rings III: Prime Ideals and Maximal Ideals; Unique Factorization; Noetherian Rings; Varieties; Grobner Bases. For all readers interested in abstract algebra.
# From Natural Numbers to Quaternions

This textbook offers an invitation to modern algebra through number systems of increasing complexity, beginning with the natural numbers and culminating with Hamilton's quaternions. Along the way, the authors carefully develop the necessary concepts and methods from abstract algebra: monoids, groups, rings, fields, and skew fields. Each chapter ends with an appendix discussing related topics from algebra and number theory, including recent developments reflecting the relevance of the material to current research. The present volume is intended for undergraduate courses in abstract algebra or elementary number theory. The inclusion of exercises with solutions also makes it suitable for self-study and accessible to anyone with an interest in modern algebra and number theory.
# A Course in Abstract Algebra 4th Edition

Designed for undergraduate and postgraduate students of mathematics the book can also be used by those preparing for various competitive examinations. The text starts with a brief introduction to results from set theory and number theory. It then goes on to cover groups, rings, vector spaces (Linear Algebra) and fields. The topics under Groups include subgroups, permutation groups, finite abelian groups, Sylow theorems, direct products, group actions, solvable and nilpotent groups. The course in Ring theory covers ideals, embedding of rings, euclidean domains, PIDs, UFDs, polynomial rings, irreducibility criteria, Noetherian rings. The section on vector spaces deals with linear transformations, inner product spaces, dual spaces, eigen spaces, diagonalizable operators etc. Under fields, algebraic extensions, splitting fields, normal and separable extensions, algebraically closed fields, Galois extensions and construction by ruler and compass are discussed. The theory has been strongly supported by numerous examples and worked out problems. There is also plenty of scope for the readers to try and solve problems on their own. NEW IN THIS EDITION • Learning Objectives and Summary with each chapter • A large number of additional worked-out problems and examples • Alternate proofs of some theorems and lemmas • Reshuffling/Rewriting of certain portions to make them more reader friendly
# A Course in Abstract Algebra 5th Edition

Designed for undergraduate and postgraduate students of mathematics, the book can also be used by those preparing for various competitive examinations. The text starts with a brief introduction to results from Set theory and Number theory. It then goes on to cover Groups, Rings, Fields and Linear Algebra. The topics under groups include subgroups, finitely generated abelian groups, group actions, solvable and nilpotent groups. The course in ring theory covers ideals, embedding of rings, Euclidean domains, PIDs, UFDs, polynomial rings, Noetherian (Artinian) rings. Topics of field include algebraic extensions, splitting fields, normal extensions, separable extensions, algebraically closed fields, Galois extensions, and construction by ruler and compass. The portion on linear algebra deals with vector spaces, linear transformations, Eigen spaces, diagonalizable operators, inner product spaces, dual spaces, operators on inner product spaces etc. The theory has been strongly supported by numerous examples and worked-out problems. There is also plenty of scope for the readers to try and solve problems on their own.New in this Edition• A full section on operators in inner product spaces.• Complete survey of finite groups of order up to 15 and Wedderburn theorem on finite division rings.• Addition of around one hundred new worked-out problems and examples.• Alternate and simpler proofs of some results.• A new section on quick recall of various useful results at the end of the book to facilitate the reader to get instant answers to tricky questions.
# A First Course in Abstract Algebra

Considered a classic by many, A First Course in Abstract Algebra is an in-depth, introductory text which gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. The Sixth Edition continues its tradition of teaching in a classical manner, while integrating field theory and new exercises.
# Rings Fields and Vector Spaces

Using the proof of the non-trisectability of an arbitrary angle as a final goal, the author develops in an easy conversational style the basics of rings, fields, and vector spaces. Originally developed as a text for an introduction to algebra course for future high-school teachers at California State University, Northridge, the focus of this book is on exposition. It would serve extremely well as a focused, one-semester introduction to abstract algebra.
# Galois Theory Third Edition

Ian Stewart's Galois Theory has been in print for 30 years. Resoundingly popular, it still serves its purpose exceedingly well. Yet mathematics education has changed considerably since 1973, when theory took precedence over examples, and the time has come to bring this presentation in line with more modern approaches. To this end, the story now begins with polynomials over the complex numbers, and the central quest is to understand when such polynomials have solutions that can be expressed by radicals. Reorganization of the material places the concrete before the abstract, thus motivating the general theory, but the substance of the book remains the same.
# Handbook of Discrete and Combinatorial Mathematics

Handbook of Discrete and Combinatorial Mathematics provides a comprehensive reference volume for mathematicians, computer scientists, engineers, as well as students and reference librarians. The material is presented so that key information can be located and used quickly and easily. Each chapter includes a glossary. Individual topics are covered in sections and subsections within chapters, each of which is organized into clearly identifiable parts: definitions, facts, and examples. Examples are provided to illustrate some of the key definitions, facts, and algorithms. Some curious and entertaining facts and puzzles are also included. Readers will also find an extensive collection of biographies. This second edition is a major revision. It includes extensive additions and updates. Since the first edition appeared in 1999, many new discoveries have been made and new areas have grown in importance, which are covered in this edition.
# A First Course in Noncommutative Rings

Aimed at the novice rather than the connoisseur and stressing the role of examples and motivation, this text is suitable not only for use in a graduate course, but also for self-study in the subject by interested graduate students. More than 400 exercises testing the understanding of the general theory in the text are included in this new edition.
# Peak Power Control in Multicarrier Communications

Peak signal power is an important factor in the implementation of multicarrier (MC) modulation schemes, like OFDM, in wireless and wireline communication systems. This 2007 book describes tools necessary for analyzing and controlling the peak-to-average power ratio in MC systems, and how these techniques are applied in practical designs. The author starts with an overview of multicarrier signals and basic tools and algorithms, before discussing properties of MC signals in detail: discrete and continuous maxima; statistical distribution of peak power; codes with constant peak-to-average power ratio are all covered, concluding with methods to decrease peak power in MC systems. Current knowledge, problems, methods and definitions are summarized using rigorous mathematics, with an overview of the tools for the engineer. The book is aimed at graduate students and researchers in electrical engineering, computer science and applied mathematics, and practitioners in the telecommunications industry.
# Introduction to the Galois Correspondence

In this presentation of the Galois correspondence, modern theories of groups and fields are used to study problems, some of which date back to the ancient Greeks. The techniques used to solve these problems, rather than the solutions themselves, are of primary importance. The ancient Greeks were concerned with constructibility problems. For example, they tried to determine if it was possible, using straightedge and compass alone, to perform any of the following tasks? (1) Double an arbitrary cube; in particular, construct a cube with volume twice that of the unit cube. (2) Trisect an arbitrary angle. (3) Square an arbitrary circle; in particular, construct a square with area 1r. (4) Construct a regular polygon with n sides for n > 2. If we define a real number c to be constructible if, and only if, the point (c, 0) can be constructed starting with the points (0,0) and (1,0), then we may show that the set of constructible numbers is a subfield of the field R of real numbers containing the field Q of rational numbers. Such a subfield is called an intermediate field of Rover Q. We may thus gain insight into the constructibility problems by studying intermediate fields of Rover Q. In chapter 4 we will show that (1) through (3) are not possible and we will determine necessary and sufficient conditions that the integer n must satisfy in order that a regular polygon with n sides be constructible.
# Rings Fields and Groups

This book provides a stimulating and unusiual introduction to the results, methods and ideas which are now commonly studied in abstract algebra courses in universities and polytechnics. The mixture of informal and formal presentation generates the enthusiasm of the reader without neglecting the axiomatic approach necessary for the serious study.
# The Tools of Mathematical Reasoning

This accessible textbook gives beginning undergraduate mathematics students a first exposure to introductory logic, proofs, sets, functions, number theory, relations, finite and infinite sets, and the foundations of analysis. The book provides students with a quick path to writing proofs and a practical collection of tools that they can use in later mathematics courses such as abstract algebra and analysis. The importance of the logical structure of a mathematical statement as a framework for finding a proof of that statement, and the proper use of variables, is an early and consistent theme used throughout the book.
# Introduction to Abstract Algebra

THIS IS A TEXTBOOK FOR STUDENTS UNDERTAKING, OR ABOUT TO UNDERTAKE, A first course in Abstract Algebra (e. g. second-year Honours students and second-or third-year Ordinary Degree students in Scottish universities; first year students in many English universities). The first edition of the book proved helpful to many such students, and it is hoped that this second edition may be found even more helpful. The main changes made in producing the se co nd edition have been the inclusion of a whole new chapter on the symmetrie group and the expansion of the existing sets of exercises, together with updating and revising of the text. The book is not an advanced treatise on group theory or on any kindred part of mathematics. But, in contrast to many elementary textbooks, it does penetrate far enough into Abstract Algebra to let the student see that the subject has worthwhile insights to offer, and to introduce hirn to some of the distinctive ways of thinking that produce interesting results. For example, in group theory, the material covered includes cyclic groups, Lagrange's theorem, homomorphisms, normal subgroups, quotient groups, and (in the new chapter) the partition of the symmetrie group of degree n into conjugacy classes and an intro duc ti on to the alternating group of degree n. Although there is only one chapter on rings (and fields), it is a lengthy chapter and covers a wide variety of ideas.
# A Concrete Introduction to Higher Algebra

This book is an informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials. The new examples and theory are built in a well-motivated fashion and made relevant by many applications - to cryptography, coding, integration, history of mathematics, and especially to elementary and computational number theory. The later chapters include expositions of Rabiin's probabilistic primality test, quadratic reciprocity, and the classification of finite fields. Over 900 exercises are found throughout the book.
# Abstract Algebra An Introduction

Abstract Algebra: An Introduction is set apart by its thematic development and organization. The chapters are organized around two themes: arithmetic and congruence. Each theme is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another. New to this edition is a groups first option that enables those who prefer to cover groups before rings to do so easily. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
# American Book Publishing Record

# Abstract Algebra An Introduction

Abstract Algebra: An Introduction is set apart by its thematic development and organization. The chapters are organized around two themes: arithmetic and congruence. Each theme is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another. New to this edition is a groups first option that enables those who prefer to cover groups before rings to do so easily. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.