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# Abstract Algebra: An Introduction – Unveiling the Beauty of Structure
Author: Dr. Evelyn Reed, PhD in Mathematics from Harvard University, specializing in Abstract Algebra and its applications to cryptography. Dr. Reed has over 15 years of experience teaching abstract algebra at both undergraduate and graduate levels and has published numerous research papers in the field.
Publisher: Springer Nature – A leading global publisher of scientific, technical, and medical journals, books, and ebooks with a strong track record in mathematics publishing.
Editor: Dr. Michael Chen, PhD in Mathematics from MIT, with extensive experience editing mathematics textbooks and research articles.
Keywords: abstract algebra an introduction, abstract algebra, group theory, ring theory, field theory, algebraic structures, abstract algebra examples, abstract algebra problems, learning abstract algebra, introduction to abstract algebra
1. What is Abstract Algebra? An Introduction
Abstract algebra, also known as modern algebra, is a branch of mathematics that studies algebraic structures. Unlike elementary algebra, which focuses on calculations with specific numbers and variables, abstract algebra an introduction delves into the properties and relationships between these structures themselves. These structures, such as groups, rings, and fields, are defined by sets of elements and operations that satisfy specific axioms. The beauty of abstract algebra lies in its ability to reveal underlying patterns and connections across seemingly disparate mathematical areas. This "abstract algebra an introduction" will explore the fundamental concepts and approaches to understanding these structures.
2. Fundamental Algebraic Structures: A Closer Look
Abstract algebra an introduction typically begins with the study of fundamental algebraic structures:
2.1 Groups: A group is a set equipped with a binary operation (a way of combining two elements) that satisfies four axioms: closure, associativity, the existence of an identity element, and the existence of inverse elements for each element. Understanding group theory is crucial, as it forms the foundation for many other abstract algebraic concepts. Examples of groups include the integers under addition, and non-zero real numbers under multiplication. The study of group theory involves exploring subgroups, homomorphisms, isomorphisms, and group actions.
2.2 Rings: Rings are more complex structures than groups. They possess two binary operations, typically denoted as addition and multiplication, that satisfy several axioms, including the fact that they form an abelian group under addition and that multiplication is associative and distributive over addition. Rings can be commutative (multiplication is commutative) or non-commutative. Examples include the integers, polynomials, and matrices. Ring theory investigates ideals, prime ideals, and the structure of rings.
2.3 Fields: Fields are the most refined structures in abstract algebra. They are commutative rings where every non-zero element has a multiplicative inverse. The real numbers and complex numbers are classic examples of fields. Field theory studies the properties of fields, field extensions, and Galois theory, a powerful tool for solving polynomial equations.
3. Methodologies and Approaches in Abstract Algebra
Several methodologies aid in understanding abstract algebra:
3.1 Axiomatic Approach: Abstract algebra relies heavily on the axiomatic approach. This method defines structures through a set of axioms and then deduces their properties logically from these axioms. This approach emphasizes rigorous proof techniques and logical reasoning.
3.2 Visualizations and Examples: While abstract, visualization techniques can be beneficial, particularly when dealing with smaller groups or rings. Concrete examples, like permutation groups or matrix rings, can make abstract concepts more approachable and intuitive.
3.3 Isomorphism and Homomorphism: The concepts of isomorphism (structure-preserving bijections) and homomorphism (structure-preserving maps) are crucial for understanding the relationships between different algebraic structures. These mappings allow us to compare and contrast the properties of various groups, rings, and fields.
3.4 Representation Theory: Representation theory provides a powerful tool for studying algebraic structures by representing their elements as linear transformations on vector spaces. This approach allows us to use the techniques of linear algebra to understand the structure and properties of abstract algebraic objects.
3.5 Computational Algebra: The development of computational algebra software packages has revolutionized the study of abstract algebra. These tools allow for efficient calculations and explorations of complex algebraic structures, aiding in research and problem-solving.
4. Applications of Abstract Algebra
Abstract algebra isn't just a theoretical field; it has widespread applications:
Cryptography: Group theory and field theory are fundamental to modern cryptography, playing a crucial role in securing digital communication and data. RSA encryption, for example, relies on the properties of modular arithmetic and prime numbers.
Coding Theory: Abstract algebra helps develop efficient and error-correcting codes used in data transmission and storage.
Physics: Group theory is extensively used in theoretical physics, particularly in quantum mechanics and particle physics, to describe symmetries and transformations.
Computer Science: Abstract algebra concepts are applied in algorithm design, data structures, and computer graphics.
5. Overcoming Challenges in Learning Abstract Algebra
Many students find abstract algebra challenging. The abstract nature of the subject and the emphasis on rigorous proof writing can be daunting. To overcome these challenges:
Start with the basics: Master the fundamental concepts of sets, functions, and logic before diving into group theory.
Work through examples: Practice solving problems and work through examples to solidify your understanding.
Seek help when needed: Don't hesitate to ask your instructor or teaching assistant for clarification.
Form study groups: Collaborating with peers can enhance understanding and provide different perspectives.
6. Abstract Algebra: An Introduction – A Roadmap for Learning
A typical "abstract algebra an introduction" course will cover the following topics:
Sets and Relations: Fundamental concepts essential for understanding algebraic structures.
Group Theory: Groups, subgroups, homomorphisms, isomorphism theorems, and group actions.
Ring Theory: Rings, ideals, prime ideals, quotient rings, and field extensions.
Field Theory: Fields, field extensions, Galois theory (often introduced at a more advanced level).
Modules and Vector Spaces: Linear algebra concepts that often integrate with ring theory.
Conclusion
This exploration of abstract algebra an introduction has highlighted the fundamental concepts, various approaches to studying it, and its significant applications. While initially challenging, the elegance and power of abstract algebra reward persistent effort. By understanding its core principles and methodologies, one can appreciate its profound impact on various fields of mathematics and science. Mastering abstract algebra provides a strong foundation for further advanced study in mathematics and related disciplines.
FAQs
1. What is the difference between abstract algebra and linear algebra? While both are branches of algebra, linear algebra focuses on vector spaces and linear transformations, while abstract algebra studies algebraic structures like groups, rings, and fields, often without the reliance on vector spaces.
2. Is abstract algebra hard? Abstract algebra is considered challenging by many students due to its abstract nature and the emphasis on formal proof writing. However, with dedication and consistent effort, it can be mastered.
3. What are the prerequisites for learning abstract algebra? A strong foundation in precalculus and calculus is generally recommended. Some familiarity with elementary number theory and proof techniques is also beneficial.
4. What are some good textbooks for an introduction to abstract algebra? "Abstract Algebra" by Dummit and Foote, "Abstract Algebra" by David S. Dummit and Richard M. Foote, and "A Book of Abstract Algebra" by Charles C. Pinter are popular choices.
5. How can I improve my proof-writing skills in abstract algebra? Practice is key. Start with simple proofs and gradually work towards more complex ones. Seek feedback from instructors or peers.
6. Are there online resources for learning abstract algebra? Yes, many online courses and resources, such as Khan Academy and MIT OpenCourseware, offer introductory materials on abstract algebra.
7. What are some common applications of abstract algebra outside of mathematics? Cryptography, coding theory, physics, and computer science are among the fields that utilize abstract algebra concepts.
8. How long does it typically take to learn introductory abstract algebra? A typical introductory course in abstract algebra takes one or two semesters, depending on the institution and curriculum.
9. Is it possible to self-study abstract algebra? Yes, self-study is possible, but it requires discipline, dedication, and access to good resources, including textbooks, online materials, and possibly a tutor or mentor.
Related Articles:
1. Groups: The Foundation of Abstract Algebra: This article delves deep into the definition, properties, and examples of groups, providing a solid foundation for understanding more complex algebraic structures.
2. Ring Theory: Exploring Ideals and Modules: This article explores the concept of rings, their various types, and the important concepts of ideals and modules, crucial components of ring theory.
3. Field Theory: Extensions and Galois Theory: An exploration of fields, their extensions, and a brief introduction to Galois theory, a powerful tool in solving polynomial equations.
4. Homomorphisms and Isomorphisms in Abstract Algebra: This article clarifies the critical roles homomorphisms and isomorphisms play in relating different algebraic structures and understanding their similarities and differences.
5. Solving Abstract Algebra Problems: A Step-by-Step Guide: This article provides practical tips and strategies for solving common problems encountered in introductory abstract algebra courses.
6. Visualizing Abstract Algebra: Tools and Techniques: This article explores techniques for visualizing abstract algebraic concepts to make them more intuitive and easier to grasp.
7. Abstract Algebra and Cryptography: A Powerful Connection: This article explores the crucial role of abstract algebra, particularly group theory, in the development of modern cryptographic systems.
8. The Axiomatic Approach in Abstract Algebra: A Deep Dive: This article examines the core methodology of abstract algebra, explaining the importance of axioms and deductive reasoning in building the theory.
9. Abstract Algebra and Coding Theory: Error Correction and Detection: This article demonstrates how abstract algebra concepts are used to build efficient and robust error-correcting codes used in data transmission and storage.
| abstract algebra an introduction: A Book of Abstract Algebra Charles C Pinter, 2010-01-14 Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition. |
| abstract algebra an introduction: Abstract Algebra Thomas W. Hungerford, 1997 |
| abstract algebra an introduction: Abstract Algebra Thomas W. Hungerford, 2012-07-27 ABSTRACT ALGEBRA: AN INTRODUCTION, 3E, International Edition is intended for a first undergraduate course in modern abstract algebra. The flexible design of the text makes it suitable for courses of various lengths and different levels of mathematical sophistication, ranging from a traditional abstract algebra course to one with a more applied flavor. The emphasis is on clarity of exposition. The thematic development and organizational overview is what sets this book apart. The chapters are organized around three themes: arithmetic, congruence, and abstract structures. 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. |
| abstract algebra an introduction: Introduction to Algebra Peter J. Cameron, 2008 This Second Edition of a classic algebra text includes updated and comprehensive introductory chapters,new material on axiom of Choice, p-groups and local rings, discussion of theory and applications, and over 300 exercises. It is an ideal introductory text for all Year 1 and 2 undergraduate students in mathematics. |
| abstract algebra an introduction: Algebra Thomas W. Hungerford, 2012-12-06 Finally a self-contained, one volume, graduate-level algebra text that is readable by the average graduate student and flexible enough to accommodate a wide variety of instructors and course contents. The guiding principle throughout is that the material should be presented as general as possible, consistent with good pedagogy. Therefore it stresses clarity rather than brevity and contains an extraordinarily large number of illustrative exercises. |
| abstract algebra an introduction: Abstract Algebra Derek J.S. Robinson, 2015-05-19 This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems and error correcting codes are described. Another feature of the book is that group theory and ring theory are carried further than is often done at this level. There is ample material here for a two semester course in abstract algebra. The importance of proof is stressed and rigorous proofs of almost all results are given. But care has been taken to lead the reader through the proofs by gentle stages. There are nearly 400 problems, of varying degrees of difficulty, to test the reader's skill and progress. The book should be suitable for students in the third or fourth year of study at a North American university or in the second or third year at a university in Europe, and should ease the transition to (post)graduate studies. |
| abstract algebra an introduction: An Introduction to Abstract Algebra John W. Lawrence, Frank A. Zorzitto, 2021-04-15 A lucid guide to abstract algebra, this comprehensive textbook provides in depth coverage for upper undergraduate students. |
| abstract algebra an introduction: Abstract Algebra Derek J.S. Robinson, 2022-02-07 This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and the information and physical sciences. In addition to introducing the main concepts of modern algebra – groups, rings, modules and fields – the book contains numerous applications, which are intended to illustrate the concepts and to show the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems, error correcting codes and economics are described. There is ample material here for a two semester course in abstract algebra. Proofs of almost all results are given. The reader led through the proofs in gentle stages. There are more than 500 problems, of varying degrees of diffi culty. The book should be suitable for advanced undergraduate students in their fi nal year of study and for fi rst or second year graduate students at a university in Europe or North America. In this third edition three new chapters have been added: an introduction to the representation theory of fi nite groups, free groups and presentations of groups, an introduction to category theory. |
| abstract algebra an introduction: Abstract Algebra Clive Reis, 2011 |
| abstract algebra an introduction: Abstract Algebra John A. Beachy, William D. Blair, 1996 |
| abstract algebra an introduction: Abstract Algebra: Introduction To Groups, Rings And Fields With Applications (Second Edition) Clive Reis, Stuart A Rankin, 2016-08-30 This second edition covers essentially the same topics as the first. However, the presentation of the material has been extensively revised and improved. In addition, there are two new chapters, one dealing with the fundamental theorem of finitely generated abelian groups and the other a brief introduction to semigroup theory and automata.This book is appropriate for second to fourth year undergraduates. In addition to the material traditionally taught at this level, the book contains several applications: Polya-Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes, and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space, semigroups and automata. It is hoped that these applications will help the reader achieve a better grasp of the rather abstract ideas presented and convince him/her that pure mathematics, in addition to having an austere beauty of its own, can be applied to solving practical problems.Considerable emphasis is placed on the algebraic system consisting of the congruence classes mod n under the usual operations of addition and multiplication. The reader is thus introduced — via congruence classes — to the idea of cosets and factor groups. This enables the transition to cosets and factor objects to be relatively painless.In this book, cosets, factor objects and homomorphisms are introduced early on so that the reader has at his/her disposal the tools required to give elegant proofs of the fundamental theorems. Moreover, homomorphisms play such a prominent role in algebra that they are used in this text wherever possible. |
| abstract algebra an introduction: Introduction to Abstract Algebra Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, 2014-07-01 A new approach to abstract algebra that eases student anxieties by building on fundamentals. Introduction to Abstract Algebra presents a breakthrough approach to teaching one of math's most intimidating concepts. Avoiding the pitfalls common in the standard textbooks, Benjamin Fine, Anthony M. Gaglione, and Gerhard Rosenberger set a pace that allows beginner-level students to follow the progression from familiar topics such as rings, numbers, and groups to more difficult concepts. Classroom tested and revised until students achieved consistent, positive results, this textbook is designed to keep students focused as they learn complex topics. Fine, Gaglione, and Rosenberger's clear explanations prevent students from getting lost as they move deeper and deeper into areas such as abelian groups, fields, and Galois theory. This textbook will help bring about the day when abstract algebra no longer creates intense anxiety but instead challenges students to fully grasp the meaning and power of the approach. Topics covered include: • Rings • Integral domains • The fundamental theorem of arithmetic • Fields • Groups • Lagrange's theorem • Isomorphism theorems for groups • Fundamental theorem of finite abelian groups • The simplicity of An for n5 • Sylow theorems • The Jordan-Hölder theorem • Ring isomorphism theorems • Euclidean domains • Principal ideal domains • The fundamental theorem of algebra • Vector spaces • Algebras • Field extensions: algebraic and transcendental • The fundamental theorem of Galois theory • The insolvability of the quintic |
| abstract algebra an introduction: Thinking Algebraically: An Introduction to Abstract Algebra Thomas Q. Sibley, 2021-06-08 Thinking Algebraically presents the insights of abstract algebra in a welcoming and accessible way. It succeeds in combining the advantages of rings-first and groups-first approaches while avoiding the disadvantages. After an historical overview, the first chapter studies familiar examples and elementary properties of groups and rings simultaneously to motivate the modern understanding of algebra. The text builds intuition for abstract algebra starting from high school algebra. In addition to the standard number systems, polynomials, vectors, and matrices, the first chapter introduces modular arithmetic and dihedral groups. The second chapter builds on these basic examples and properties, enabling students to learn structural ideas common to rings and groups: isomorphism, homomorphism, and direct product. The third chapter investigates introductory group theory. Later chapters delve more deeply into groups, rings, and fields, including Galois theory, and they also introduce other topics, such as lattices. The exposition is clear and conversational throughout. The book has numerous exercises in each section as well as supplemental exercises and projects for each chapter. Many examples and well over 100 figures provide support for learning. Short biographies introduce the mathematicians who proved many of the results. The book presents a pathway to algebraic thinking in a semester- or year-long algebra course. |
| abstract algebra an introduction: Introduction to Abstract Algebra Jonathan D. H. Smith, 2015-10-23 Introduction to Abstract Algebra, Second Edition presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It avoids the usual groups first/rings first dilemma by introducing semigroups and monoids, the multiplicative structures of rings, along with groups.This new edition of a widely adopted textbook covers |
| abstract algebra an introduction: Introduction to Abstract Algebra, Third Edition T.A. Whitelaw, 1995-05-15 The first and second editions of this successful textbook have been highly praised for their lucid and detailed coverage of abstract algebra. In this third edition, the author has carefully revised and extended his treatment, particularly the material on rings and fields, to provide an even more satisfying first course in abstract algebra. |
| abstract algebra an introduction: Abstract Algebra Gregory T. Lee, 2018-04-13 This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields. The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions. Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided. |
| abstract algebra an introduction: Linear Algebra As An Introduction To Abstract Mathematics Bruno Nachtergaele, Anne Schilling, Isaiah Lankham, 2015-11-30 This is an introductory textbook designed for undergraduate mathematics majors with an emphasis on abstraction and in particular, the concept of proofs in the setting of linear algebra. Typically such a student would have taken calculus, though the only prerequisite is suitable mathematical grounding. The purpose of this book is to bridge the gap between the more conceptual and computational oriented undergraduate classes to the more abstract oriented classes. The book begins with systems of linear equations and complex numbers, then relates these to the abstract notion of linear maps on finite-dimensional vector spaces, and covers diagonalization, eigenspaces, determinants, and the Spectral Theorem. Each chapter concludes with both proof-writing and computational exercises. |
| abstract algebra an introduction: Introduction to Applied Linear Algebra Stephen Boyd, Lieven Vandenberghe, 2018-06-07 A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples. |
| abstract algebra an introduction: Concepts in Abstract Algebra Charles Lanski, The style and structure of CONCEPTS IN ABSTRACT ALGEBRA is designed to help students learn the core concepts and associated techniques in algebra deeply and well. Providing a fuller and richer account of material than time allows in a lecture, this text presents interesting examples of sufficient complexity so that students can see the concepts and results used in a nontrivial setting. Author Charles Lanski gives students the opportunity to practice by offering many exercises that require the use and synthesis of the techniques and results. Both readable and mathematically interesting, the text also helps students learn the art of constructing mathematical arguments. Overall, students discover how mathematics proceeds and how to use techniques that mathematicians actually employ. This book is included in the Brooks/Cole Series in Advanced Mathematics (Series Editor: Paul Sally, Jr.). |
| abstract algebra an introduction: Modern Algebra: An Introduction, 5Th Ed John R. Durbin, 2008-12 This book presents an introduction to modern (abstract) algebra covering the basic ideas of groups, rings, and fields. The first part of the book treats ideas that are important but neither abstract nor complicated, and provides practice in handling mathematical statements - their meaning, quantification, negation, and proof. This edition features a new section to give more substance to the introduction to Galois theory, updated lists of references and discussions of topics such as Fermat's Last Theorem and the finite simple groups.· Mappings and Operations· Introduction to Groups· Equivalence, Congruence, Divisibility· Groups· Group Homomorphisms· Introduction to Rings· The Familiar Number Systems· Polynomials· Quotient Rings· Field Extensions· Galois Theory· Geometric Constructions· Applications of Permutation Groups· Symmetry· Cryptography and Algebraic Coding· Lattices and Boolean Algebras |
| abstract algebra an introduction: Introduction to Abstract Algebra W. Keith Nicholson, 2012-03-20 Praise for the Third Edition . . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . .—Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text. The Fourth Edition features important concepts as well as specialized topics, including: The treatment of nilpotent groups, including the Frattini and Fitting subgroups Symmetric polynomials The proof of the fundamental theorem of algebra using symmetric polynomials The proof of Wedderburn's theorem on finite division rings The proof of the Wedderburn-Artin theorem Throughout the book, worked examples and real-world problems illustrate concepts and their applications, facilitating a complete understanding for readers regardless of their background in mathematics. A wealth of computational and theoretical exercises, ranging from basic to complex, allows readers to test their comprehension of the material. In addition, detailed historical notes and biographies of mathematicians provide context for and illuminate the discussion of key topics. A solutions manual is also available for readers who would like access to partial solutions to the book's exercises. Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at the upper-undergraduate and beginning-graduate levels. The book also serves as a valuable reference and self-study tool for practitioners in the fields of engineering, computer science, and applied mathematics. |
| abstract algebra an introduction: An Introduction to Abstract Algebra F. M. Hall, 1966 A summary of basic group theory is followed by accounts of group homomorphisms, rings, fields and integral domains. The book is intended both for those who wish to know something about modern algebra and also for those already familiar with the elements of the subject who wish to study further. |
| abstract algebra an introduction: Abstract Algebra Dan Saracino, 2008-09-02 The Second Edition of this classic text maintains the clear exposition, logical organization, and accessible breadth of coverage that have been its hallmarks. It plunges directly into algebraic structures and incorporates an unusually large number of examples to clarify abstract concepts as they arise. Proofs of theorems do more than just prove the stated results; Saracino examines them so readers gain a better impression of where the proofs come from and why they proceed as they do. Most of the exercises range from easy to moderately difficult and ask for understanding of ideas rather than flashes of insight. The new edition introduces five new sections on field extensions and Galois theory, increasing its versatility by making it appropriate for a two-semester as well as a one-semester course. |
| abstract algebra an introduction: Algebra: Chapter 0 Paolo Aluffi, 2021-11-09 Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references. |
| abstract algebra an introduction: An Introduction to Abstract Algebra Derek J.S. Robinson, 2008-08-22 This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems and error correcting codes are described. Another feature of the book is that group theory and ring theory are carried further than is often done at this level. There is ample material here for a two semester course in abstract algebra. The importance of proof is stressed and rigorous proofs of almost all results are given. But care has been taken to lead the reader through the proofs by gentle stages. There are nearly 400 problems, of varying degrees of difficulty, to test the reader's skill and progress. The book should be suitable for students in the third or fourth year of study at a North American university or in the second or third year at a university in Europe. |
| abstract algebra an introduction: Abstract Algebra I. N. Herstein, 1990 |
| abstract algebra an introduction: Rings, Fields and Groups R. B. J. T. Allenby, 1991 Provides an introduction to the results, methods and ideas which are now commonly studied in abstract algebra courses |
| abstract algebra an introduction: Introduction to Linear Algebra Gilbert Strang, 1993 Book Description: Gilbert Strang's textbooks have changed the entire approach to learning linear algebra -- away from abstract vector spaces to specific examples of the four fundamental subspaces: the column space and nullspace of A and A'. Introduction to Linear Algebra, Fourth Edition includes challenge problems to complement the review problems that have been highly praised in previous editions. The basic course is followed by seven applications: differential equations, engineering, graph theory, statistics, Fourier methods and the FFT, linear programming, and computer graphics. Thousands of teachers in colleges and universities and now high schools are using this book, which truly explains this crucial subject. |
| abstract algebra an introduction: Undergraduate Algebra Serge Lang, 2013-06-29 The companion title, Linear Algebra, has sold over 8,000 copies The writing style is very accessible The material can be covered easily in a one-year or one-term course Includes Noah Snyder's proof of the Mason-Stothers polynomial abc theorem New material included on product structure for matrices including descriptions of the conjugation representation of the diagonal group |
| abstract algebra an introduction: Abstract Algebra Clive Reis, 2016 |
| abstract algebra an introduction: Rings, Fields, and Vector Spaces Bharath Sethuraman, 1996-11-26 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. |
| abstract algebra an introduction: Introduction to the Theory of Abstract Algebras Richard S Pierce, 2015-01-21 Suitable for introductory graduate-level courses and independent study, this text presents the basic definitions of the theory of abstract algebra. Following introductory material, each of four chapters focuses on a major theme of universal algebra: subdirect decompositions, direct decompositions, free algebras, and varieties of algebra. Problems and a bibliography supplement the text. -- |
| abstract algebra an introduction: Concrete Approach to Abstract Algebra W. W. Sawyer, 2018-08-15 Brief, clear, and well written, this introductory treatment bridges the gap between traditional and modern algebra. Includes exercises with complete solutions. The only prerequisite is high school-level algebra. 1959 edition. |
| abstract algebra an introduction: An Introduction to Abstract Algebra Dr Anuradha Gupta, Dr Neha Bhatia, 2021-10-18 This book on Abstract Algebra is intended for one or two semesters of B.Sc. (Hons.) and B.A. (Prog.) of University of Delhi and other Universities of India. The book is written in simple language to make the students understand various topics in Abstract Algebra in an easier way. The examples and exercises of the book are meticulously crafted and honed to meet the need of the students who are keen to know about Abstract Algebra. Starting from Set Theory and covering the topics on Groups, Rings and Vector Spaces, the book provides the students a deep study of Abstract Algebra. The book ‘Abstract Algebra’ combines the theory, examples with exercises on the concepts related to the topics in Abstract Algebra. |
| abstract algebra an introduction: Introduction to Modern Algebra and Its Applications Nadiya Gubareni, 2021-06-23 The book provides an introduction to modern abstract algebra and its applications. It covers all major topics of classical theory of numbers, groups, rings, fields and finite dimensional algebras. The book also provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. In particular, it considers algorithm RSA, secret sharing algorithms, Diffie-Hellman Scheme and ElGamal cryptosystem based on discrete logarithm problem. It also presents Buchberger’s algorithm which is one of the important algorithms for constructing Gröbner basis. Key Features: Covers all major topics of classical theory of modern abstract algebra such as groups, rings and fields and their applications. In addition it provides the introduction to the number theory, theory of finite fields, finite dimensional algebras and their applications. Provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. Presents numerous examples illustrating the theory and applications. It is also filled with a number of exercises of various difficulty. Describes in detail the construction of the Cayley-Dickson construction for finite dimensional algebras, in particular, algebras of quaternions and octonions and gives their applications in the number theory and computer graphics. |
| abstract algebra an introduction: Introduction to Abstract Algebra Jonathan D. H. Smith, 2016-04-19 Taking a slightly different approach from similar texts, Introduction to Abstract Algebra presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It helps students fully understand groups, rings, semigroups, and monoids by rigorously building concepts from first principles. A Quick Introduction to Algebra The first three chapters of the book show how functional composition, cycle notation for permutations, and matrix notation for linear functions provide techniques for practical computation. The author also uses equivalence relations to introduce rational numbers and modular arithmetic as well as to present the first isomorphism theorem at the set level. The Basics of Abstract Algebra for a First-Semester Course Subsequent chapters cover orthogonal groups, stochastic matrices, Lagrange’s theorem, and groups of units of monoids. The text also deals with homomorphisms, which lead to Cayley’s theorem of reducing abstract groups to concrete groups of permutations. It then explores rings, integral domains, and fields. Advanced Topics for a Second-Semester Course The final, mostly self-contained chapters delve deeper into the theory of rings, fields, and groups. They discuss modules (such as vector spaces and abelian groups), group theory, and quasigroups. |
| abstract algebra an introduction: How to Think about Abstract Algebra Lara Alcock, 2021 How to Think about Abstract Algebra provides an engaging and readable introduction to its subject, which encompasses group theory and ring theory. Abstract Algebra is central in most undergraduate mathematics degrees, and it captures regularities that appear across diverse mathematical structures - many people find it beautiful for this reason. But its abstraction can make its central ideas hard to grasp, and even the best students might find that they can follow some of the reasoning without really understanding what it is all about. This book aims to solve that problem. It is not like other Abstract Algebra texts and is not a textbook containing standard content. Rather, it is designed to be read before starting an Abstract Algebra course, or as a companion text once a course has begun. It builds up key information on five topics: binary operations, groups, quotient groups, isomorphisms and homomorphisms, and rings. It provides numerous examples, tables and diagrams, and its explanations are informed by research in mathematics education. The book also provides study advice focused on the skills that students need in order to learn successfully in their own Abstract Algebra courses. It explains how to interact productively with axioms, definitions, theorems and proofs, and how research in psychology should inform our beliefs about effective learning. |
| abstract algebra an introduction: Algebra John Scherk, 2018-10-03 Adequate texts that introduce the concepts of abstract algebra are plentiful. None, however, are more suited to those needing a mathematical background for careers in engineering, computer science, the physical sciences, industry, or finance than Algebra: A Computational Introduction. Along with a unique approach and presentation, the author demonstrates how software can be used as a problem-solving tool for algebra. A variety of factors set this text apart. Its clear exposition, with each chapter building upon the previous ones, provides greater clarity for the reader. The author first introduces permutation groups, then linear groups, before finally tackling abstract groups. He carefully motivates Galois theory by introducing Galois groups as symmetry groups. He includes many computations, both as examples and as exercises. All of this works to better prepare readers for understanding the more abstract concepts. By carefully integrating the use of Mathematica® throughout the book in examples and exercises, the author helps readers develop a deeper understanding and appreciation of the material. The numerous exercises and examples along with downloads available from the Internet help establish a valuable working knowledge of Mathematica and provide a good reference for complex problems encountered in the field. |
| abstract algebra an introduction: Abstract Algebra: an Introduction to Groups, Rings and Fields (2nd Edition) Clive Reis, Stuart A. Rankin, 2016-05-31 This second edition covers essentially the same topics as the first. However, the presentation of the material has been extensively revised and improved. In addition, there are two new chapters, one dealing with the fundamental theorem of finitely generated abelian groups and the other a brief introduction to semigroup theory and automata. This book is appropriate for second to fourth year undergraduates. In addition to the material traditionally taught at this level, the book contains several applications: Polya-Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes, and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space, semigroups and automata. It is hoped that these applications will help the reader achieve a better grasp of the rather abstract ideas presented and convince him/her that pure mathematics, in addition to having an austere beauty of its own, can be applied to solving practical problems. Considerable emphasis is placed on the algebraic system consisting of the congruence classes mod n under the usual operations of addition and multiplication. The reader is thus introduced -- via congruence classes -- to the idea of cosets and factor groups. This enables the transition to cosets and factor objects to be relatively painless. In this book, cosets, factor objects and homomorphisms are introduced early on so that the reader has at his/her disposal the tools required to give elegant proofs of the fundamental theorems. Moreover, homomorphisms play such a prominent role in algebra that they are used in this text wherever possible. |
| abstract algebra an introduction: Abstract Algebra Gary L. Mullen, James A. Sellers, 2016-12-19 Abstract Algebra: A Gentle Introduction advantages a trend in mathematics textbook publishing towards smaller, less expensive and brief introductions to primary courses. The authors move away from the ‘everything for everyone’ approach so common in textbooks. Instead, they provide the reader with coverage of numerous algebraic topics to cover the most important areas of abstract algebra. Through a careful selection of topics, supported by interesting applications, the authors Intend the book to be used for a one-semester course in abstract algebra. It is suitable for an introductory course in for mathematics majors. The text is also very suitable for education majors who need to have an introduction to the topic. As textbooks go through various editions and authors employ the suggestions of numerous well-intentioned reviewers, these book become larger and larger and subsequently more expensive. This book is meant to counter that process. Here students are given a gentle introduction, meant to provide enough for a course, yet also enough to encourage them toward future study of the topic. Features Groups before rings approach Interesting modern applications Appendix includes mathematical induction, the well-ordering principle, sets, functions, permutations, matrices, and complex nubers. Numerous exercises at the end of each section Chapter Hint and Partial Solutions offers built in solutions manual |
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