Preface. Birkhoff & Mac Lane’s Algebra is a brilliant book. I should probably spend some time with it again, actually. Also, I apologize for such a. In Garrett Birkhoff and Saunders Mac Lane published A Survey of Modern Algebra. The book became a classic undergraduate text. Below we examine a. Garrett BirkhoffHarvard University Saunders Mac Lane The University of Chicago A SURVEY OF ern fourth.

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But I’d sure keep one on reserve in the library for my students to browse. This book also does not match a clear unorthodox vision of how the subject should be taught, unlike, say, Aluffi or Adkins.

Preliminary Thoughts

Sign up using Facebook. Adding in some applications may be good, too; I do not remember many being in Mac Lane.

I don’t think hitting students with category theory unless they’ve already had considerable exposure to algebra is a good idea. But there is no dearth of good reference works in algebra, and in the reviewer’s birkhofff the present textbook will prove more useful than another encyclopedic treatise would have been. For undergraduates, we obviously need to ensure they see examples, learn the basic theory, be able to write proofs, etc.

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

Hungerford is a common alternative I did not discuss, but I have never found it very exciting. We enjoyed teaching and writing algebra because it was clear, exciting, and fun to present. Modern algebra prospered mightily in the decadesfrom functional analysis to algebraic geometry – not to mention our own respective researches on lattices and on categories. Such students are strongly advised to do supplementary reading, or at least browsing, in the references listed at the end.

Because of the authors’ emphasis on “method” rather than “fact” the book will not be of much use as a reference work. A very striking feature of the book is its broad point of view. Birkhoff and Mac Lane also want to teach their students to prove things, of course. Similarly, books like Cohn, Grillet, and Jacobson can be too advanced or too focused on being references.

Modern algebra also enables one to reinterpret the results of classical algebra, giving them far greater unity and generality. Interesting historical references appear in a number of places.

A typical first undergraduate brikhoff may cover group theory through the isomorphism theorems and the structure theorem for finite abelian groups,possibly including group actions and the Sylow theorems c. They do not avoid using universal properties, and they do not always bother to give students something concrete to hold on.


Also, your review page is quite comprehensive! As Mac Lane did years ago, it is best to supplement the text with something easier, something intended for undergraduates. I am hard pressed to put a single book above the others, however. We must not risk them never learning it. Anx 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.

Rotman may be a good primary text. Certainly, one can survive albebra, but it is probably suboptimal for most. Perhaps a lecture reviewing elementary set theoretical notions, then cover some mcalane algebra introducing basic category theory after seeing direct sumsthen cover some ring theory, then plenty of group theory, then modules and advanced linear algebra, followed by field and Galois theory, representation theory using algebras and specializing quickly to groupscommutative algebra including some applications to algebraic geometry and the likeand finally homological algebra, with some advanced or extra topics at the end, if possible e.

This will get everyone to do the work they will need to do to read many papers, which are not uniformly well written and often take having a mature audience as an excuse not to motivate or show any work. Our Survey presented an exciting mix of classical, axiomatic, and conceptual ideas about algebra at a time when this combination was new. 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 algeebra Waerden, and Philip Hall.

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. Chapter IIIand some field and vector space theory c.

Survey of Modern Algebra

The authors are quick to indicate applications and careful to motivate and illustrate abstractions. Our collaboration involved some compromises.


Probably the best way to appreciate the vitality and growth of mathematics today is to study modern algebra. Motivation, examples, clear writing, reasonable exercises, they are all there. I do not think using algebras is a problem, but three whole chapters is probably a bit much, so some would have to be cut. There is also contact with the field of mathematical logic in the chapter on the algebra of classes and with the ideas maclabe topology in the proof of the fundamental theorem of algebra.

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. 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. I think graduate courses should use category theory pretty openly.

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. 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.

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.

Although some additions and rearrangements have been made for this edition, the content remains essentially the same [as the edition]. Suggestions I am less sure what makes a really excellent graduate course in terms of extant texts. The most important parts of each theory are included and that is all that can be asked of an introductory textbook.

Maflane is a great supplementary text, however. Those desiring a text replete with possibilities for courses tailored to various kinds of students should welcome this new edition.