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He returned to Harvard inthe year after I had given a course in modern algebra on the undergraduate level for the first time. Although my course was well attended, I was much more research-oriented than teaching-oriented. Mac Lane had had much more teaching experience than I, and I think the popularity of our book owes more to him than to me. His problems and his organization of linear algebra were especially timely. Our collaboration involved some compromises. When I taught “modern algebra” in “Math 6” the first time, inI began with sets and ended with groups.

The next year Mac Lane put group theory first, and set theory Boolean algebra last! That was characteristic of his freshness, his initiative, and his lack of respect for conformity; but it came as a slight shock to me at the time.

After teaching the course again the next year, I suggested that we co-author a book, usable by our colleagues, so that we wouldn’t have to alternate teaching it forever, and he agreed.

One of us would draft a chapter and the other would revise it. The longer chapters are his; the shorter ones mine. I had taught algebra courses at Harvard when I was an instructor, and at Cornell I taught algebra out of the book by Bocher; at Chicago, out of a book, ‘Modern Higher Algebra’ by Albert; and at Harvard again out of my own notes. I had taught out of most of the extant books. I knew how it should be done and so did Garrett. He had been doing the same thing.

We got together and, inwe wlgebra ‘A Survey of Modern Algebra’. It was the standard textbook for undergraduate courses in modern algebra. 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.

We had the good fortune to write a book birlhoff the subject at the right time. 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. In writing the present text we have endeavoured to set forth this formal or “abstract” approach, but we have been guided by a much broader interpretation of the significance of modern algebra.

Much of this significance, it seems to us, lies in the imaginative appeal of the subject. Accordingly, we have tried throughout to express the conceptual background of the various definitions used. We have done this by illustrating each new term agebra as many familiar examples as possible. This seems especially important in an elementary text because it bigkhoff to emphasize the fact that the birkjoff concepts all arise from the analysis of concrete situations.

To develop the student’s power to think for himself in terms of the new concepts, we have included a wide maclame of exercises on each topic. Some of these exercises are computational, some explore further examples mwclane the new concepts, and others give additional theoretical developments.

Exercises of the latter type serve the important function ,aclane familiarizing the student with the construction of a formal algebrra.

The selection of exercises is sufficient to allow an instructor to adapt the text to students of quite varied degrees of maturity, of undergraduate or first year graduate level. Modern algebra also enables one to reinterpret the results of classical algebra, algebrs them far greater unity and generality.

Therefore, instead of omitting these results, we have attempted to incorporate them systematically within the framework of the ideas of modern algebra. We have also tried not to bifkhoff sight of the fact that, for many students, the value of algebra lies in its applications to other fields: 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 orthogonal 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.

In this book Professors Birkhoff and Mac Lane have made an important contribution to the pedagogy of algebra. Their emphasis is on the methods and spirit of modern algebra rather than on the subject matter for itself.

The word “survey” in the title is quite accurate; for, although many topics are treated, none of them is really completely developed. The most important parts of each theory are included and that is all that can be asked of an introductory textbook.

Because of the authors’ emphasis on “method” rather than “fact” the book will not be of much use as a reference work. 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 bitkhoff.

The authors express the belief that “for many students, the value of algebra lies in its applications to other fields: This is a text on modern algebra that is particularly suited for a first year graduate course or for an advanced undergraduate course.

### Survey of Modern Algebra

A very striking feature of the book is its broad point of view. There are contacts with many branches of mathematics and so it can serve as an introduction to nearly the whole of modern mathematics.

Thus there is a careful development of birmhoff numbers, such as Dedekind cuts, borkhoff such set-theoretic concepts as order, countability and cardinal number are discussed. Throughout the study of matrices and quadratic forms the geometric point of view is emphasized.

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 algsbra fundamental theorem of algebra. This exposition of the elements of modern algebra has been planned with great skill, and the plan has been carried through very successfully.

It is a unified and comprehensive introduction to modern algebra. The classical algebra is nicely embedded in this structure, as are also applications to other fields of thought. This book is distinguished by its procedure from the concrete to the birkhoft.

Familiar examples are carefully presented to illustrate each new term or idea which is introduced. Then the abstract definition appears simple, and the theoretical properties which are deduced from the definition exhibit the power of the concept. This book is distinguished also by the great clarity with which all details have been presented. The rejuvenation of algebra by the systematic use of the postulational method and the ideas and point of view of abstract group theory has been one birkgoff the crowning achievements of twentieth century mathematics.

Although many of the basic results stem back to Kronecker, Amclane and Steinitz, the present-day subject is largely the creation of the great woman mathematician, Emmy Noether. 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.

The size and plan of the book preclude a comprehensive treatment of any one topic; in compensation, the authors are able to say something about nearly every important topic, and they usually succeed in saying the really important things.

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. We hope that the present book will prove an adequate reference for those wishing to apply the basic concepts of modern algebra to other branches of mathematics, physics, and statistics. Maclanf also hope it will give a solid introduction to this fascinating and rapidly birkhodf subject, to those students interested in modern algebra for its own sake.

Such students are strongly advised to do supplementary reading, or at least browsing, in the references listed at the end. Only in this birkhoff will they be able to appreciate the full richness of the subject.

In preparing the revised edition we have added several important topics equations of stable type, dual spaces, the projective group, the Jordan and rational canonical forms for matrices, etc. Numerous additional exercises, summarising useful formulas and facts, have been included.

Some material, aogebra that on linear algebra, has been rearranged in the light of experience. Although some additions and rearrangements have been made for this edition, the content remains essentially the same [as the edition]. This well-known textbook has served, in the last twelve years, to introduce a great many students to the fundamental concepts of modern algebra in an extraordinarily effective way.

Alvebra 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.

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.

The ratio of definitions to theorems and exercises is kept low. Interesting historical macoane appear in a number of places. The authors are to be congratulated on having improved an already excellent text. Occasionally a textbook makes possible a real leap forward in ways mwclane learning. Inwhen the first edition of this book appeared, the curriculum in algebra was the result of a hodge-podge accumulation.

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. It provided a synthesis formerly obtainable only after much more advanced work. 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.

This reviewer can testify to its appeal to students. The revised edition differs only in minor rearrangements and additions. For the social birlhoff whose mathematical studies have virkhoff through the calculus, this book can confidently be urged as the thing to study next.

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 the material and to do the problems.

One of the best things about this book is the balanced approach to rigour and abstractness in relation to intuitive appreciation and concrete application. The authors are quick to indicate applications and careful to motivate and illustrate abstractions.

The present edition represents a refinement of an already highly useful text. The original comprehensive Survey has been reordered somewhat and augmented to the extent of approximately fifty pages. Only the last five chapters remain unchanged.

## Preliminary Thoughts

Instructors who have used the original edition with college classes appreciate its scope. Those desiring a text replete with possibilities for courses tailored maclanw various kinds of students should welcome this new edition. Teachers of mathematics in secondary schools may want this book in their personal libraries.

Probably the best way to appreciate the vitality and growth of mathematics today is to study modern algebra. Nowhere can teachers better jaclane today’s spirit of mathematics. Moreover, many of the examples in this text might help teachers to communicate this spirit to their students. For over twenty years this text has been the “classic” work in its field.

In the new third edition, the authors have modernized and improved the material in many details. 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.

This third edition of a standard text alyebra modern algebra is substantially the same as the revised edition [of ]. A section on bilinear forms and tensor products has been added to the chapter 7 on vector spaces, while Chapter 11, macclane entitled “Boolean algebras and lattices”, contains a new introduction to Boolean algebras, as well as a section on the representation of such by sets.

Beyond this, occasional sections have been revised and a few problems have been added to some of the exercises. Chapters give an introduction to the theory of linear and polynomial equations in commutative rings.