Algebra: Abstract and Concrete

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9781616100100: Algebra: Abstract and Concrete
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For a two-semester, undergraduate- level course in Introduction to Modern/Abstract Algebra.This innovative and versatile text starts by counting symmetries and then quickly gets into group theory. Symmetry (and geometry) is a theme throughout. Author has website as well.

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Addresses the needs of those who often have weak analytic ability, little idea of strategies of problem solving, and less of the patience and confidence needed to work proofs. It introduces readers to the active practice of mathematics — luring them away from the view of mathematics as a system of rules and procedures and helping them learn to think things through for themselves and to explain accurately what they have understood. Uses linear algebra and complex numbers throughout. Highlights all theorems (propositions, lemmas, corollaries) and definitions in boxes. Features an abundance of high-quality exercises.

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This text is an introduction to "modern" or "abstract" algebra for undergraduate students. The book addresses the conventional topics; groups, rings, and fields, with symmetry as a unifying theme. This subject matter is central and ubiquitous in modern mathematics and in applications ranging from quantum physics to digital communications.

The most important goal of this book is to engage students in the active practice of mathematics. Students are given the opportunity to participate and investigate, starting on the first page. Exercises are plentiful, and working exercises should be the heart of the course.

This text provides a thorough introduction to abstract algebra at a level suitable for upper-level undergraduates. The text would also be useful for an undergraduate topics course (for example, on geometric aspects of group theory or on Galois theory).

The required background for using this text is a standard first course in linear algebra. I have included a brief summary of linear algebra in an appendix to help students review. I have also provided appendices on sets, logic, mathematical induction, and complex numbers. The instructor may wish to go through this "corequisite" material systematically, or to dip into it from time to time as needed. It might also be useful to recommend a short supplementary text on set theory, logic, and proofs to be used as a reference and aid; several such texts are currently available.

The text is adaptable to different teaching styles. My own preference is increasingly to lecture little, and to use class time for discussing problems. But those who wish to present the material in systematic lectures will find the subject matter cleanly organized and presented in the text. I offer one piece of pedagogical advice both to instructors and students: This stuff takes time, give it the time it needs. Students have an enormous amount to learn "between the lines" of the text. They not only have to learn the mathematics, but they need to learn how mathematics is done, how to read and write mathematics, and how to approach solving problems. They need to learn to tinker, to try examples, to formulate and solve a simpler problem, to try a special case, to think about analogies, to guess at intermediate results, and so on. It is important for the instructor to slow the pace of the course in order to encourage exploration, and it is important for the student to devote the (many) hours which are actually needed to absorb the material and to solve the problems.

The subject treated in this text is usually called abstract algebra. In common language, abstract means both "difficult" and "impractical," and it is a little unfortunate to start out by labeling the subject as hard but useless! It won't be out of place here to make some (encouraging) remarks about abstraction. It tikes some effort to remember that even the counting numbers were once (and in principle still are) an enormous abstraction. But they are familiar, they no longer seem difficult, and no one would doubt their usefulness. Abstractions with which we have become familiar eventually lose their aura of abstractness, but those with which we are not yet familiar seem abstract indeed. So it is with the ideas of this course. They may seem abstract today, but as they become familiar they will seem more concrete.

Mathematics involves a continual interplay between the abstract and the concrete: Abstraction is necessary in order to understand concrete phenomena, and concrete phenomena are necessary in order to understand the abstractions. Meanwhile, as one continues to study mathematics the perceived boundary between the abstract and concrete inevitably shifts.

Organization.

The text has been substantially revised for this second edition. The most important innovation is the long introductory chapter, entitled "Algebraic Themes," which treats many of the more elementary topics—symmetries, the integers, modular arithmetic, polynomials, and permutations—and introduces the important algebraic objects—groups, rings and fields—that are the subject of the rest of the book. The chapter also contains new sections on counting and cryptography. Chapter 6, on rings has been expanded, with a more complete discussion of Euclidean, principal ideal, and unique factorization domains. The classification of finite abelian groups has been moved to Chapter 3, on products of groups, and a section on vector spaces and bases has, appropriately, been added to this chapter. There is a new appendix containing a review of linear algebra (Appendix E).

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