- Author:
- W. Keith Nicholson
Praise for the Third Edition". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."Zentralblatt MATHThe 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 subgroupsSymmetric polynomialsThe proof of the fundamental theorem of algebra using symmetric polynomialsThe proof of Wedderburn's theorem on finite division ringsThe proof of the Wedderburn-Artin theoremThroughout 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.
Index Terms
- Introduction to Abstract Algebra
Recommendations
Edgar R. Chavez
In writing this book, Nicholson set for himself a rather ambitious goal: to write a self-contained introductory book that could be used in a variety of settings for a variety of audiences. By and large, he succeeded. The book is primarily intended for a two-semester course for undergraduates or a one-semester course at the graduate level. The book contains the material normally expected in an abstract algebra course-groups, rings, Galois theory, and so on-and it is developed using a rigorous axiomatic method. But Nicholson also includes additional topics, optional sections, application-related material, and historical notes, so the book can be geared to mathematics, computer science, or engineering students. Finally, Nicholson keeps in mind the needs of readers who might want to use the book for self-study or reference. The tone is consistently clear, the pace is measured, and the methods are often illustrated before they are formalized in theorems. The subject matter is developed carefully and thoroughly, emphasizing readability and comprehension. Particularly toward the beginning, examples are plentiful, and these are revisited throughout the book. These help with motivation and soften the load of understanding the abstract nature of the material. Exercises are also plentiful, many of them of a computational nature, intended to develop familiarity with and comprehension of the material. By consistently working through the exercises, the reader can deepen his understanding and appreciation of the power and beauty of the concepts and notation presented. Interestingly, Nicholson does not use problems to sneak in material that will be required later on. All of the conceptual material is developed in the text, and none in the exercises. The book includes hints or solutions for selected exercises, and a separate solutions manual is available from the publisher. To make the book self-contained, Nicholson uses two schemes. In chapter 0, "Preliminaries," he covers proofs, sets, mappings, and equivalences; at the end, he includes four appendices covering complex numbers, matrix algebra, Zorn's lemma, and a proof of the recursion theorem. In principle, then, the reader can find here all he needs; in practice, the student with scant acquaintance with the axiomatic method and formal proofs will have a rude awakening. Ideally, the reader will already possess that rare quality usually referred to as mathematical maturity, something that students in advanced classes are expected to have acquired. This implicit requirement is somewhat ameliorated by Nicholson's careful use of proof techniques in examples before using them in proofs. The book also includes a list of the notation used in the text, a recommended but not exhaustive bibliography, and an excellent index. On a rather captious note, I would have preferred critical concepts to be introduced in sections clearly marked as "Definitions" rather than having them included in the descriptive text. This would have made the book easier to use for review and reference. However, that is not the way Nicholson chose to emphasize readability. This is a large book, with 15 pages of prefatory material and 535 pages of text. Clearly, instructors can tailor several courses by selecting chapters and sections. My feeling is that this book is better suited for a full two-semester sequence rather than a one-semester course. That would give students a better opportunity to fully absorb the material, peruse the numerous examples, and work through a significant number of exercises. This could also be an excellent adjunct to more theoretically oriented textbooks used in more intensive courses. Online Computing Reviews Service
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