A BOOK OF ABSTRACT ALGEBRA : Second Edition
FOR UPSC, STATE PCS, SSC, BANKINGS, CAT, NDA, CDS AND OTHER NATIONAL & STATE LEVEL COMPETITIVE EXAMS
Once, when I was a student struggling to understand modern algebra, I was told to view this subject as an intellectual chess game, with conventional moves and prescribed rules of play. I was ill served by this bit of extemporaneous advice, and vowed never to perpetuate the falsehood that mathematics is purely—or primarily—a formalism. My pledge has strongly influenced the shape and style of this book. While giving due emphasis to the deductive aspect of modern algebra, I have endeavored here to present modern algebra as a lively branch of mathematics, having considerable imaginative appeal and resting on some firm, clear, and familiar intuitions. I have devoted a great deal of attention to bringing out the meaningfulness of algebraic concepts, by tracing these concepts to their origins in classical algebra and at the same time exploring their connections with other parts of mathematics, especially geometry, number theory, and aspects of computation and equation solving. In an introductory chapter entitled Why Abstract Algebra?, as well as in numerous historical asides, concepts of abstract algebra are traced to the historic context in which they arose. I have attempted to show that they arose without artifice, as a natural response to particular needs, in the course of a natural process of evolution. Furthermore, I have endeavored to bring to light, explicitly, the intuitive content of the algebraic concepts used in this book. Concepts are more meaningful to students when the students are able to represent those concepts in their minds by clear and familiar mental images. Accordingly, the process of concrete concept-formation is developed with care throughout this book. I have deliberately avoided a rigid conventional format, with its succession of definition, theorem, proof, corollary, example. In my experience, that kind of format encourages some students to believe that mathematical concepts have a merely conventional character, and may encourage rote memorization. Instead, each chapter has the form of a discussion with the student, with the accent on explaining and motivating. In an effort to avoid fragmentation of the subject matter into loosely related definitions and results, each chapter is built around a central theme and remains anchored to this focal point.
In the later chapters especially, this focal point is a specific application or use. Details of every topic are then woven into the general discussion, so as to keep a natural flow of ideas running through each chapter. The arrangement of topics is designed to avoid tedious proofs and long-winded explanations. Routine arguments are worked into the discussion whenever this seems natural and appropriate, and proofs to theorems are seldom more than a few lines long. (There are, of course, a few exceptions to this.) Elementary background material is filled in as it is needed. For example, a brief chapter on functions precedes the discussion of permutation groups, and a chapter on equivalence relations and partitions paves the way for Lagrange’s theorem. This book addresses itself especially to the average student, to enable him or her to learn and understand as much algebra as possible. In scope and subject-matter coverage, it is no different from many other standard texts. It begins with the promise of demonstrating the unsolvability of the quin tic and ends with that promise fulfilled. Standard topics are discussed in their usual order, and many advanced and peripheral subjects are introduced in the exercises, accompanied by ample instruction and commentary. I have included a copious supply of exercises—probably more exercises than in other books at this level. They are designed to offer a wide range of experiences to students at different levels of ability. There is some novelty in the way the exercises are organized: at the end of each chapter, the exercises are grouped into exercise sets, each set containing about six to eight exercises and headed by a descriptive title. Each set touches upon an idea or skill covered in the chapter.
About book
Book Name: A book of abstract algebra
Publication: Dover Publications, Inc., Mineola, New York
Authors: Charles C. Pinter
Language: English
Pages: 358
Size: 7mb
Format: pdf
Uploaded: Google drive
A BOOK OF ABSTRACT ALGEBRA
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