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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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A Survey of Modern Algebra

Familiar examples are carefully presented to illustrate each new term or idea which is introduced. Our Survey presented an exciting mix of classical, axiomatic, and conceptual ideas maclaje algebra at a time when this combination was new. Interesting historical references appear in a number of places.

When I taught “modern algebra” in “Math 6” the first time, inI began with sets and ended with groups. The authors are to be congratulated on birhkoff improved an already excellent text. Because of the authors’ emphasis on “method” rather than “fact” the book will not be of much use as a reference work. We have done this by illustrating each new term by as many familiar examples as possible.

The ratio of definitions to theorems and exercises is kept low. But they want to teach them algebra even more. Exercises of the latter type serve the important function of familiarizing the student with the construction of a formal proof. Our collaboration involved some compromises. The next year Mac Lane put group theory first, and set theory Boolean algebra last!

The most striking characteristic of modern algebra is the deduction of the theoretical properties of such formal systems as groups, rings, fields, and vector spaces. While it can be used as a reference, it should rather be read through carefully over a period of years – one must think in terms of years if one wishes to absorb fully malcane material and to do the problems. Although two or three books on the new algebra have already appeared in English, the present volume appears to the reviewer to be the best all-round introduction to the subject, unique in its clarity, balance, generality and inclusiveness.

Only in this way will they be able to appreciate the full richness of the subject. Some of these exercises are computational, some explore further examples of the new concepts, and others give additional theoretical developments.

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

This book is distinguished by its procedure from the concrete to the abstract. Chapter 6 introduces noncommutative algebra through its simplest and most fundamental concept: Still other arrangements are possible. Their emphasis is on the methods and spirit of modern algebra rather than on the subject matter for itself.


After explaining the conceptual content of the classical theory of equations, our book tried to bring out the connections of newer algebraic concepts with geometry and analysis, connections that had indeed inspired many of these concepts in the first place.

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His paper, “On the Structure of Abstract Algebras” founded a new branch of mathematics, universal algebra. We hope that our book will continue to serve not only as a text but also as a convenient reference for those wishing to apply the basic concepts of modern algebra to other branches of mathematics, including statistics and computing, and also to physics, chemistry, and engineering. All instructor resources are now available on our Instructor Hub.

Chapters 4 and 5 develop the basic algebraic properties of the real and complex fields which are of such paramount importance for geometry and physics. It does this by discussing examples of mathematical systems or situations already partially familiar to the student, isolating important properties of these as postulates, and deducing some of the consequences of these postulates.

Adopting this as the main textbook for an undergraduate abstract algebra course would today be an eccentric move. I knew how it should be done and so did Garrett.

Terminology and notation which has become outmoded since the Revised Edition was published in have been brought up-to-date; material on Boolean algebra and lattices has been completely rewritten; an introduction to tensor products has been added; numerous problems have been replaced and many new ones added; and throughout the book are hundreds of minor revisions to keep the work in the forefront of modern algebra literature and pedagogy. But the authors’ delight birohoff what was then a new subject shines through their writing, and their willingness to be informal when necessary was a smart move.

One of us would draft a chapter and the other would revise it. The rejuvenation of algebra by the systematic use of the postulational method and the ideas and point of albebra of abstract group theory has been one of the crowning achievements of twentieth century mathematics.

These responsibilities were in effect combined in our activity. This has influenced us in our emphasis on the real and complex fields, on groups of transformations as contrasted with abstract groups, on symmetric matrices and reduction to diagonal form, on the classification of quadratic forms under the biirkhoff and Euclidean groups, and finally, in the inclusion of Boolean algebra, lattice theory, and transfinite numbers, all of which are important in mathematical logic and in the modern theory of real functions.


Many algebra textbooks are so concerned about the process of learning to prove things that they communicate a sense of algebar subject as forbidding and stiff, dedicated to formalism and brkhoff. Of course, the book came partly from England through Garrett, who had been influenced by Philip Hall when he worked with him at Cambridge England.

There are contacts with many branches of mathematics and so it can serve as an introduction to nearly the whole of modern mathematics. Chapters give an introduction to the theory of linear and polynomial equations in commutative rings.

His problems and his organization of linear algebra were especially timely. Accordingly, we have tried throughout to express the conceptual background of the various definitions used. There is also contact with the field of mathematical logic in the chapter on the algebra of classes and with the ideas of topology in the proof of the fundamental theorem of algebra.

Therefore, instead of omitting these results, we have attempted to incorporate them systematically within the framework of the ideas of modern algebra. Numerous additional exercises, summarising useful formulas and facts, have been included.

A survey of modern algebra / by Garrett Birkhoff and Saunders MacLane – Details – Trove

This is followed in Chapter 12 by a brief discussion of transfinite numbers. It began to sell well as soon as the war was over, in at about – annually. For over twenty years this text has been the “classic” work in its field. 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.

The original comprehensive Survey has been reordered somewhat and augmented to the extent of approximately fifty pages. We provide complimentary e-inspection copies of primary textbooks to instructors considering our books for course adoption. Request an e-inspection copy. In writing the present text we have endeavoured to set forth this formal or algrbra approach, but we have been guided by a much broader interpretation of the significance of modern algebra.

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.