A survey of modern algebra. Bibliography: p. Includes index. 1. Algebra, Abstract. I. MacLane, Saunders, (date) joint author. II. Title. QAB57 “This classic text introduces abstract algebra using familiar and concrete examples that illustrate each concept as it is presented. It covers such topics as the role. A survey of modern algebra / by Garrett Birkhoff and Saunders MacLane Birkhoff, articles: Garrett Birkhoff, Greatest common divisor, Saunders Mac Lane.

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Then the abstract definition appears simple, and the theoretical properties which are deduced from the definition exhibit the power of the concept.

A Survey of Modern Algebra – Garrett Birkhoff, Saunders Mac Lane – Google Books

But there is no dearth of good reference works in algebra, and in the reviewer’s opinion the present textbook will prove more useful than another encyclopedic treatise would have been. It provided a clear and enthusiastic emphasis on the then new modern and axiomatic view of algebra, as advocated by Emmy Noether, Emil Artin, van der Waerden, and Philip Hall. The present edition represents a refinement of an already highly useful text. He had been doing the same thing. Byrelatively new concepts inspired by it had begun to influence homology theory, operator theory, the theory of topological groups, and many other domains of mathematics.

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I knew how it should be done and so did Garrett. Already read this title? Chapter 6 introduces noncommutative algebra through its simplest and most fundamental concept: We aimed to combine the abstract ideas with suitable emphasis on examples and illustrations. He received the nation’s highest award for scientific achievement, the National Medal of Science, in The classical algebra is nicely embedded in this alegbra, as are also applications to other fields of thought.


Saunders Mac Lane was the author or co-author of more than research papers and six books. I had taught out of most of the extant books.

Algenra additional exercises, summarising useful formulas and facts, have been included. Some of these exercises are computational, some explore further examples of the new concepts, and others give additional theoretical developments.

These theorems are then applied to some familiar and to some less familiar examples, thus broadening the student’s viewpoint without getting him lost in abstractions. Chapters give an introduction to the theory of linear and polynomial equations in commutative rings.

A Survey of Modern Algebra

This is a text on modern algebra that is particularly suited for a first year graduate course or for an advanced undergraduate course. Nevertheless, it is still a book well worth reading. A section on bilinear forms and tensor products has been added to the chapter 7 on vector spaces, while Chapter 11, now entitled “Boolean algebras and lattices”, contains a new introduction to Boolean algebras, as well as a section on the representation of such by sets.

This third edition of a standard text on modern algebra is substantially the same as the revised maclan [of ]. A famous mathematician once remarked to me that abd he knew who had worked through A Survey of Modern Algebra had come to love the subject. Moreover, it was written in a clear and enthusiastic style that conveyed to the reader an appreciation of the aesthetic character of the subject as well as its rigour and power. It was the standard textbook for undergraduate courses in modern algebra.

Here care is taken to keep in the foreground the fundamental role played by algebra in Euclidean, affine, and projective geometry.

One of us would draft a chapter and the other would revise it. He spent most of his career as a professor of mathematics at Harvard University.

This independence is intended to make the book useful not only for a full-year course, assuming only high-school algebra, but also for various shorter courses. That may overstate things, since my friend probably knows more mathematicians than students who got fed up and left. In addition the book is enlivened by striking applications of modern algebra to other branches of science and made eminently teachable by the inclusion of numerous excellent problems and exercises.


Both are excellent books I have called this book Advanced Modern Algebra in homage to thembut times have changed since their first appearance: A ‘Survey of Modern Algebra’ made it possible to teach an undergraduate course that reflected the richness, vigour, and unity of the subject as it is growing today.

Only in this way will they be able to appreciate the full richness of the subject.

Throughout the study of matrices and quadratic forms the geometric point of view is emphasized. Nowhere can teachers better catch today’s spirit of mathematics. The familiar domain of integers and the rational field are emphasized, together with the rings of integers modulo n and associated polynomial rings.

It could be through conference attendance, group discussion or directed reading to name just a few examples.

Probably the best way to appreciate the vitality and growth of mathematics today is to study modern algebra. To develop the student’s power to think for himself in terms of the new concepts, we have included a wide variety of exercises on each topic.

Instructors who have used the original edition with college classes appreciate its scope. It is a unified and comprehensive introduction to modern algebra.