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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Here care is taken to keep in the foreground the fundamental role played by algebra in Euclidean, affine, and projective geometry.

The ratio of definitions to theorems and exercises is kept low. In fact, I think this book works better as anr supplement. But, hey, at least there are lattices!

He returned to Harvard inthe year after I had given a course in modern algebra on the undergraduate level for the first time. The most impressive thing about Rowen is how up to date it is. Also, the mathematics department at Harvard both emphasized research and expected all faculty members to be steadily active in teaching undergraduates. While I’m really enjoying the book,I’ve got a real dilemma gnawing at me.

It’s a projected book to accompany my new publishing company to instruct penniless students on low-cost mathematics sources for self study. It has a lot of linear algebra, which is good, and it is not too hard, but it requires some work.

I mentioned flaws of a few books in my answer. Aluffi is really something special, but it is probably a bit too category theory-centric for students new to the subject in the sense that they will not have enough familiarity with the lower-level stuff. The word “survey” in the title is quite accurate; for, although many topics are treated, none of them is really completely developed.

Because of it’s unorthodox structure and old fashioned style,though,I’m not sure I’d use the book for a graduate course. The Mathemagician 2 6. For example, a semester or quarter course covering linear algebra may be based on Chaptersthe real and complex fields being emphasized.

At both the graduate and undergraduate level, it seems there are books that are better in parts or in whole. Certainly, one can survive it, but it is probably suboptimal for most. 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 book is distinguished by its procedure from the concrete to the abstract. Therefore, instead of omitting these results, we have attempted to incorporate them systematically within the framework of the ideas of modern algebra.

Also, your review page is quite comprehensive! Garrett and I combined our preliminary notes to publish with MacMillan in our joint book, Survey of Modern Algebra.

Dummit and Foote is great for group theory, but it suffers in other parts agebra may also too big and encyclopedic. Chapter VIImore field theory and linear algebra c. Our third edition, infinally included tensor products of vector spaces, while the fourth edition clarified the treatment of Boolean algebras and lattices. We enjoyed teaching and writing algebra because it was clear, exciting, and fun to present.

Students of the first type would benefit from a more challenging text, whereas students of the second type would benefit from an easier one. Would like to hear feedback on this question.

Yet, I also think a course should start with basic material. It is a unified and comprehensive introduction to modern algebra. I also think there should probably be more commutative algebra.

Math Forum Discussions

Logic suggests the standard groups, rings, fields, modules, vector spaces, etc. The most important parts of each theory are included and that is all that can be asked of an introductory textbook. Those desiring a text replete with possibilities for courses tailored to various kinds of students should welcome algebraa new edition. Dual spaces and tensor products are also discussed, but generalizations to modules over rings are not considered.

Our book, first published 50 years ago, was intended snd present this exciting new view of algebra to American undergraduate and beginning graduate students. The Basic Graduate Year includes some good applications to algebraic geometry and algebraic number theory, for example. Post as a guest Name. 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.

Home Questions Macllane Users Unanswered.

Preliminary Thoughts

After teaching the course again the next year, I suggested that we co-author a book, usable by our akgebra, so that we wouldn’t have to alternate teaching it forever, and he agreed. Throughout the study of matrices and quadratic forms the geometric point of view is emphasized. I think these principles in lecture and text, along with hard work by the students, lots of student-student communication including some forced working togethersome student-professor msclane including some forced communication, e.


You do not define a group as a groupoid with one object, at least not in an introductory course. That the basic objects of groups and mmaclane are presented early on and come back to several times, maclaen than spending one-hundred pages only talking about groups, then changing mxclane only talking about rings, should also help students see how algebraic structures are related; it also makes the beginning a,gebra, and provides an opportunity for students to save themselves if they fall behind early on.

I think you could cover the entire book, minus perhaps Chapter XIV on lattices, which are not typically emphasized and are partly historical, and possibly Chapter VIII, which is a bit odd, in my opinion. Finally, the last three chapters provide an introduction to general commutative algebra and arithmetic: Only in this way will they be able to appreciate the full richness of the subject.

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 algerba the forefront of modern algebra literature and pedagogy.

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.

I am joking, if it is not clear. Although not enough to reconcile this problem, their exposition on categorical notions is quite clear, at least, albeit maybe not as good as some treatments that have since come. Being able to handle abstraction does not mean students should not learn many concrete, basic examples, however, nor does it mean they should learn things at the most abstract possible level and be expected to figure out the less abstract consequences on their own.

Birkhoff and Mac Lane also want to teach their students to prove things, of course. Accordingly, we have tried throughout to express the conceptual background of the various definitions used.