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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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. There are contacts with many branches of mathematics and so it can serve as an introduction to nearly the whole of modern mathematics.

Chapter 6 introduces noncommutative algebra through its simplest and most fundamental concept: The authors are to be congratulated on having improved an already excellent text. 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 do not think using algebras is a problem, but three whole chapters is probably a bit much, so some would have to be cut. But at the same time, since the book contains very little homological algebra and no representation theory ibrkhoff the definition,can it really be used as graduate course today? We got together and, inwe published ‘A Survey of Modern Algebra’.

We must not risk them never learning it.

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

In both undergraduate courses and graduate courses students’ abilities and background are quite variable; that much has not changed in the past few decades. Rotman’s Advanced Modern Algebra is pretty light on the Galois theory but he gets points for having a chapter on algebras.

A very striking feature of the book is its broad point of view. Mac Lane was elected to the National Academy of Sciences in Birkhoff and Mac Lane’s book first appeared inand van der Waerden’s book first appeared in This is followed in Chapter 12 by a brief discussion of transfinite numbers. We aimed to combine the abstract ideas with suitable emphasis on examples and illustrations.


Our book, first published 50 years ago, was intended to present this exciting new view of algebra to American undergraduate and beginning graduate students.

In this way, I do not expect today’s students are substantially better than past students, either. But, hey, at least there are lattices!

Logic suggests the standard groups, rings, fields, modules, vector spaces, etc. Potential Courses I think the text could be used for either undergraduate or graduate students with decent results, but I do not think it is the best choice as a primary text. Garrett and I combined our preliminary notes to publish with MacMillan in our joint book, Survey of Modern Algebra.

I understand from professors who were students at Harvard that MacLane himself would use the book-in all it’s versions,from the first mimeographed drafts in the early ‘s to the 3rd edition he used in his last teaching days in the mid’s-for both undergraduate and graduate courses in algebra depending on the strength of the students, which would vary enormously from class to class.

Chapter VIImore field theory and linear algebra c. In we algfbra a carefully polished second edition, in which polynomials over general fields were treated before specializing to the real field. For undergraduates, we obviously need to ensure they see examples, learn maclwne basic theory, be able to write proofs, etc.

Supplementing this with another book to get some coverage of representation theory and homological algebra is probably ideal. 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. The truth, though, is that undergraduates are fairly unlikely to read their textbook.


You mention Herstein and Vinberg, which are good; Artin or Gallian also work, for example.

Preliminary Thoughts

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. They do not avoid using universal properties, and they do not always bother to give students something concrete to hold on.

Then the abstract definition appears simple, and the theoretical properties which are deduced from the definition zlgebra the power algebrx the concept. Lots of people cannot stand Lang’s writing, too. Again, eventually the students should be given this stuff to them.

The “Modem Algebra” of our title refers to the conceptual and axiomatic approach to this subject initiated by David Hilbert a century ago. Nowhere can teachers better catch today’s spirit of mathematics. I agree, I do not believe Rowen is all that fitting for a standard or nearly standard course.

We have websites like Math Stack Exchange where we can ask users for opinions, but it would be nice to have many more bibliographies of mathematical subjects than there currently are. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The most important parts of each theory are included and that is all that can be asked of an introductory textbook.

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.