My father [George David Birkhoff] recognized Mac Lane's exceptional qualities, and got him invited as a Benjamin Peirce Instructor in 1934-1936. He returned to Harvard in 1938, the 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. My conservative inclusion of material from the then traditional "college algebra" and "theory of equations" courses (Bôcher, Dickson, Fine) may have helped with its initial success, as did my recognition that Galois theory used vector spaces.
Our collaboration involved some compromises. When I taught "modern algebra" in "Math 6" the first time, in 1937-38, I 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, in 1941, we published 'A Survey of Modern Algebra'. It was the standard textbook for undergraduate courses in modern algebra. There weren't any other books that really represented the modern spirit that came from Göttingen; van der Waerden's didn't exist then in translation. 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 on 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 by as many familiar examples as possible. This seems especially important in an elementary text because it serves to emphasize the fact that the abstract 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 variety of exercises on each topic. Some of these exercises are computational, some explore further examples of the new concepts, and others give additional theoretical developments. Exercises of the latter type serve the important function of familiarizing the student with the construction of a formal proof. 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, giving 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 lose sight of the fact that, for many students, the value of algebra lies in its applications to other fields: higher analysis, geometry, physics, and philosophy. 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 been. ... The authors express the belief that "for many students, the value of algebra lies in its applications to other fields: higher analysis, geometry, physics, and philosophy." This belief is reflected in many places throughout the book; but one of the most striking examples is found in their treatment of matrices and determinants, which is so arranged as to be an excellent foundation for a later study of Hilbert and Banach spaces.

This is a text on modern algebra that is particularly suited for a first year graduate course or for an advanced undergraduate course. 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 real numbers, such as Dedekind cuts, and 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 the 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 abstract. 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 of the crowning achievements of twentieth century mathematics. Although many of the basic results stem back to Kronecker, Dedekind and Steinitz, the present-day subject is largely the creation of the great woman mathematician, Emmy Noether. "Modern Algebra," by one of her pupils, B L van der Waerden, will always remain the classical account of the subject as she conceived it. 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. ... the book is emphatically recommended either as a text, an introduction to the literature or a bird's-eye view of one of the great branches of modern mathematics.

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. We also hope it will give a solid introduction to this fascinating and rapidly growing 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 way 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, especially 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 1941 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. 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. 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 references 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 of learning. In 1941, when 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 scientist whose mathematical studies have reached 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. Instructors who have used the original edition with college classes appreciate its scope. Those desiring a text replete with possibilities for courses tailored to 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 catch 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 1953 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 on modern algebra is substantially the same as the revised edition [of 1953]. 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. Beyond this, occasional sections have been revised and a few problems have been added to some of the exercises.

Chapters 1-3 give an introduction to the theory of linear and polynomial equations in commutative rings. The familiar domain of integers and the rational field are emphasized, together with the rings of integers modulo n and associated polynomial rings. 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. Chapter 6 introduces noncommutative algebra through its simplest and most fundamental concept: that of a group. The group concept is applied systematically in Chapters 7-10, on vector spaces and matrices. Here care is taken to keep in the foreground the fundamental role played by algebra in Euclidean, affine, and projective geometry. Dual spaces and tensor products are also discussed, but generalizations to modules over rings are not considered. Chapter 11 includes a completely revised introduction to Boolean algebra and lattice theory. This is followed in Chapter 12 by a brief discussion of transfinite numbers. Finally, the last three chapters provide an introduction to general commutative algebra and arithmetic: ideals and quotient-rings, extensions of fields, algebraic numbers and their factorization, and Galois theory. Many of the chapters are independent of one another; for example, the chapter on group theory may be introduced just after Chapter 1, while the material on ideals and fields (§§13.1 and 14.1) may be studied immediately after the chapter on vector spaces. 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. For example, a semester or quarter course covering linear algebra may be based on Chapters 6-10, the real and complex fields being emphasized. A semester course on abstract algebra could deal with Chapters 1-3, 6-8, 11, 13, and 14. Still other arrangements are possible. 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.

The "Modern Algebra" of our title refers to the conceptual and axiomatic approach to this subject initiated by David Hilbert a century ago. This approach, which crystallized earlier insights of Cayley, Frobenius, Kronecker, and Dedekind, blossomed in Germany in the 1920s. By 1930, relatively new concepts inspired by it had begun to influence homology theory, operator theory, the theory of topological groups, and many other domains of mathematics. Our book, first published 50 years ago, was intended to present this exciting new view of algebra to American undergraduate and beginning graduate students. We had tried out our somewhat differing ideas of how this should be done in a course at Harvard for three successive years, before reorganizing and presenting them in textbook form. 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. ... Modern algebra prospered mightily in the decades 1930-1960, from functional analysis to algebraic geometry - not to mention our own respective researches on lattices and on categories. Our Survey presented an exciting mix of classical, axiomatic, and conceptual ideas about algebra at a time when this combination was new. It began to sell well as soon as the war was over, in 1948-53 at about 2000 - 3000 annually. In 1951-53 we prepared a carefully polished second edition, in which polynomials over general fields were treated before specializing to the real field. Other more minor changes and additions helped to increase its popularity, with annual sales in the range 14,000-15,000. Our third edition, in 1965, finally included tensor products of vector spaces, while the fourth (1977) edition clarified the treatment of Boolean algebras and lattices. Our Survey in 1941 presented an exciting mix of classical and conceptual ideas about algebra. These ideas are still most relevant and worthy of enthusiastic presentation. They embody the elegance, precision, and generality which are the hallmark of mathematics!

Garrett and I combined our preliminary notes to publish with MacMillan in 1941 our joint book, Survey of Modern Algebra. 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. We aimed to combine the abstract ideas with suitable emphasis on examples and illustrations. ... We enjoyed teaching and writing algebra because it was clear, exciting, and fun to present. The book was prepared at a time when both of us were assistant professors, so without tenure. Yes, we did know then that research mattered for tenure, but our joy in teaching was somehow connected with our respective research. Also, the mathematics department at Harvard both emphasized research and expected all faculty members to be steadily active in teaching undergraduates. These responsibilities were in effect combined in our activity. Then and later we took part in the flow of new ideas from discovery to use and to present to students.

When I was a student, Birkhoff and Mac Lane's A Survey of Modern Algebra was the text for my first algebra course, and van der Waerden's Modern Algebra was the text for my second course. Both are excellent books (I have called this book Advanced Modern Algebra in homage to them), but times have changed since their first appearance: Birkhoff and Mac Lane's book first appeared in 1941, and van der Waerden's book first appeared in 1930. There are today major directions that either did not exist over 60 years ago, or that were not then recognized to be so important. These new directions involve algebraic geometry, computers, homology, and representations (A Survey of Modern Algebra has been rewritten as Mac Lane-Birkhoff, Algebra, Macmillan, New York, 1967, and this version introduces categorical methods; category theory emerged from algebraic topology, but was then used by Grothendieck to revolutionize algebraic geometry).

Garrett Birkhoff published more than 200 papers and supervised more than 50 Ph.Ds. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences. He spent most of his career as a professor of mathematics at Harvard University. During the 1930s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders Mac Lane, substantially advanced American teaching and research in abstract algebra. His 1935 paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics, universal algebra. Saunders Mac Lane was the author or co-author of more than 100 research papers and six books. Mac Lane was elected to the National Academy of Sciences in 1949. He received the nation's highest award for scientific achievement, the National Medal of Science, in 1989. Mac Lane received two Guggenheim Fellowships and visited Australia as a Fulbright Scholar. Other honours include both the Chauvenet Prize and the Distinguished Service award of the Mathematical Association of America, the Steele Career Prize of the American Mathematical Society, and honorary fellowship in the Royal Society of Edinburgh.

Adopting this as the main textbook for an undergraduate abstract algebra course would today be an eccentric move. Nevertheless, it is still a book well worth reading. I would certainly place it in the hands of an interested undergraduate wondering what algebra was all about, particularly one who had already taken linear algebra. A famous mathematician once remarked to me that everyone he knew who had worked through A Survey of Modern Algebra had come to love the subject. That may overstate things, since my friend probably knows more mathematicians than students who got fed up and left. But the authors' delight in what was then a new subject shines through their writing, and their willingness to be informal when necessary was a smart move. Many algebra textbooks are so concerned about the process of learning to prove things that they communicate a sense of the subject as forbidding and stiff, dedicated to formalism and precision. Birkhoff and Mac Lane also want to teach their students to prove things, of course. But they want to teach them algebra even more.