You know that the study of algebraic structures that includes groups, rings, fields, modules, vector spaces, lattices, and algebras is called abstract algebra also known as modern algebra.
It is, in fact, about patterns. You will see that one pattern repeating itself across mathematics and you try to extract the essential elements of that pattern and turn them into a definition. This process gives you groups, rings, fields, vector spaces, etc.
Abstract algebra is potentially useful, but in order to get any benefit from it, you need to study as many different examples as you can, do calculations, use calculations with matrices, permutations, symmetries, etc, think geometrically and draw pictures. In doing so, you essentially need to seriously study the best Abstract Algebra books.
Best Abstract Algebra Books
1. A Book of Abstract Algebra: (Dover Books on Mathematics)
This classic abstract algebra book covers extensively all of the basic topics of a typical course in elementary abstract algebra. The book approaches an easy-to-read text that offers you an intuitive approach. Most importantly, it features informal discussions followed by thematically arranged exercises. Though it is primarily intended for Undergraduate Courses in abstract algebra, it is suitable for Junior and Senior-Level Math majors and future math teachers.
You will get an introductory chapter that traces concepts of abstract algebra from their historical roots. The next chapters avoid the conventional format of definition-theorem-proof-corollary-example; instead, they take the form of a discussion with students, focusing on explanations and offering motivation. Each chapter is based on a central theme, usually a specific application.
It breaks down the most essential algebraic topic in a simple way so you can understand the topic by yourself. The author provided core reasons why you need to study abstract algebra, operations on a set, the definition of groups, elementary properties of groups, subgroups, etc.
It makes Functions, Groups of Permutations, the Concept of Isomorphism in mathematics, Cyclic Groups, Counting Cosets, Homomorphisms, Quotient Groups. Rings Definitions and Elementary Properties, etc easy to learn for you.
Moreover, in order to make you master abstract algebra, the book addresses Integral Domains, the Integers, Factoring into Primes, Elements of Number Theory, the Rings of Polynomials, Substitution in Polynomial, solving equations by radicals, and other basic topics.
Essentially, the author provides you with an elementary background as needed and discusses standard topics in their usual order. Also, you will find many advanced and peripheral subjects in the plentiful exercises, which are accompanied by ample instruction and commentary offer a wide range of experiences to students at different levels of ability.
2. Abstract Algebra, 3rd Edition
This Abstract Algebra book by David S. Dummit and Richard M. Foote is a widely accepted Introduction to Abstract Algebra book that effectively helps students experience the power and beauty that develops from the rich interplay between different areas of mathematics.
In fact, the textbook carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results. It uses numerous examples and exercises to help you practically understand the subject deeply. The book approaches in a way so that students can gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings. Therefore, it is regarded as one of the best abstract algebra books for you.
The Abstract Algebra text is developed for a full-year introduction to abstract algebra at the Advanced Undergraduate and Graduate Level. But it contains potentially more topics than would normally be covered in one year. The book covers carefully Algebraic Number Theory, Algebraic Topology, Algebraic Geometry, Representation Theory, Lie Groups, etc.
The book presents comprehensively the basics of abstract algebra into six parts. Part one named Group Theory covers an Introduction to Groups, Subgroups, Quotient Groups, Homomorphisms, Action Groups, Direct and Semidirect Products, and Abelian Groups.
Part 2 introduces the Ring Theory, Introduction to Rings, Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains, Polynomial Rings, etc.
Then the master book addresses Modules and Vector Spaces, Introduction to Module Theory, Modules over Principal Ideal Domains.
You will get an easy interpretation of Field Theory and Galois Theory, the basic Theory of Field Extension, the Fundamental Theorem of Galois Theory in part four.
The book will address you with an Introduction to Communicative Rings, Algebraic Geometry, Homological Algebra, Radicals, and Affine Varieties, the Prime Spectrum of a Ring, and more. And in last part of the book, discusses deeply the Representation Theory and Character Theory.
3. Abstract Algebra: An Introduction, 3rd Edition
The book especially is designed for a first Undergraduate course in modern abstract algebra. The most flexible design of the great book 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 textbook is organized around two basic themes: arithmetic and congruence. Each theme is developed first for the Integers, then for Polynomials, and finally for Rings and Groups. So students can see where many abstract concepts of algebra come from, why they are important, and how they relate to one another.
It features a groups-first option that enables students who want to cover groups before rings to do so easily. And most essentially, proofs for beginners in the early chapters in the book, which are broken into steps, each of which is explained and proved in detail. Therefore, it would be a great algebra learning tool for you at first by yourself.
It is well organized in the Core Course covers the topic Arithmetic in Z, Congruence in Z, Rings, Arithmetic in F(x), Congruence in F(x), and Congruence-Class Arithmetic, Ideals, and Quotient Rings, Groups, Normal Subgroups, and Quotient Groups, etc.
Advances Topics like Group Theory, Direct Products, the Structure of Finite Groups, Arithmetic in Integral Domains, Principal Ideal Domains and Unique, Field Extensions, and Galois Theory in detail so that you can grasp the concept easily.
In the final part of the book, you will get Excursions and Applications, Public-Key Cryptography, the Chinese Remainder Theorem, Applications of the Chinese Remainder Theorem, Geometric Constructions, Algebraic Coding Theory, Linear Codes in an easy to understand approach to help you achieve command over them.
4. Abstract Algebra for Beginners: A Rigorous Introduction
If you are starting abstract algebra from the scratch and intend to gain skills in this subject, Abstract Algebra for Beginners: A Rigorous Introduction to Groups, Rings, Fields, Vector Spaces, Modules, Substructures, Homomorphisms, Quotients, Group Actions, Polynomials, and Galois Theory would be a perfect text for you. The book consists of a series of basic to intermediate lessons in abstract algebra. Most importantly, the book features all the proof-writing skills that are essential for advanced study in mathematics.
Abstract Algebra for Beginners is perfect for teachers and professors teaching an Undergraduate Course or basic Graduate Course in abstract algebra, high school teachers working with advanced math students, and students wishing to see the type of mathematics they would be exposed to as a math major.
The book offers 16 lessons consisting of Basic to Intermediate topics in abstract algebra. A problem set after each lesson is arranged by difficulty level. In addition, a complete solution guide is included as a downloadable PDF file.
You will get a thorough and comprehensive study of the most important subjects in Abstract Algebra. Though it is an introductory textbook, it is at the same time goes very deep explaining topics clearly. It will be definitely the best abstract algebra book for you as its theory, examples, and exercises are very well distributed throughout the work highly effective for teaching and learning.
The potential book comprehensively explains 16 basic and most essential lessons, for example, Sets and Subsets, Algebraic Structures, Relations and Partitions, Functions and Equinumerosity, Number Systems and Induction, Substructures, Homomorphisms and Isomorphisms, Number Theory, Number Theoretic Applications, Quotients, Structure Theorems, Permutations and Determinants, Sylow Theory and Group Actions, Polynomials, Field Theory, Galois Theory.
5. Abstract Algebra: A Student-Friendly Approach
When you are studying abstract algebra independently or facing problems understanding your professor, this book is absolutely for you. Abstract Algebra: A Student-Friendly Approach not only makes learning abstract algebra so easy for you but also makes you think mathematically and do mathematics while reading the book. The book covers all the basic and traditional topics in an introductory abstract algebra course.
Its only prerequisite is high school algebra.
The textbook is an extremely effective Introduction to Abstract Algebra. The claim that the only prerequisite is high school algebra is right. The authors define the symbols of mathematical logic such as ⊂, ⊃, ∼, ≁, ∃, ∈, ∋, ⇒, and ⇔, explaining them both verbally and by the use of truth tables. It presents clearly and concisely the underlying logic of abstract algebra.
At the end of each chapter, it provides not only student exercises but also interspersed throughout the text as each new concept is introduced.
Abstract Algebra: A Student-Friendly Approach covers Sets, Mathematical Statements, Compound Statements, Proof Techniques, Operations on Sets, Relations, and Functions, Binarry Operations, Introduction to Groups, Symmetric Group of Degree n, Divisibility Properties of the Integers, Equivalence Relations, Congruence, Symmetries of a Regular Polygon, Subgroups, Isomorphic Groups, and Cyclic Groups, Coset Decomposition, and Lagrange’s Theorem, Quotient Groups, Group Homomorphisms, Introduction to Rings, Subrings and Quotient Rings, Ring Homomorphisms, Integral Domains and Fields, Polynomials, Vector Spaces and more.
The most effective section of this book is that each chapter comes with Review Questions, Answers to the Review Questions, and Homework Questions that would be highly helpful to learn each topic clearly and precisely indeed.