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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By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies. 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 ibrkhoff.

I think it would definitely be too difficult for any but the very best undergraduates and I think bkrkhoff are birkhocf books that would be better for that purpose: Those desiring a text replete with possibilities for courses tailored to various kinds of students should welcome this new edition. What level students today can Algebra be used for? I had taught out of most of the extant books.

They do not avoid using universal properties, and they do not always bother to give students something malcane to hold on. A wonderful and detailed response.

Instructors who have used the original edition with college classes appreciate its scope.

## Preliminary Thoughts

Birjhoff think too much about algebra pedagogy and textbooks. I am less sure what makes a really excellent graduate course in terms of extant texts. 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.

Students of the first type would benefit from a more challenging text, whereas students of the second type would benefit from an easier one. 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 algebbra has been added; numerous problems have been replaced and mac,ane 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.

The longer chapters are his; the shorter ones mine. This book birkhiff distinguished also by the great clarity with which all details have been presented.

Maximum generality entailing sophisticated machinery can seem efficient in the abstract, but it rarely works so well in practice. Young students can handle abstraction. Hungerford’s Algebra is a pretty good book, but the author includes little about homological algebra, and the only time you see the word “representation” is when discussing category theory.

They embody the elegance, precision, and generality which are the hallmark of mathematics! Rowen has been talked about a good bit, which is deserving for its extensive presentation on algebras and many applications, but I am not sure starting with modules is a good idea, for example. 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.

A semester course on abstract algebra could deal with Chapters, 11, 13, and 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. Our Survey in presented an exciting mix of classical and conceptual ideas about algebra.

There should also be specific portions of book and lecture that are intentionally plain, definition-theorem-proof type things, but where all the richness is introduced later down the line, as to not snub the students. Then the abstract definition appears simple, and the theoretical properties which are deduced from the definition exhibit the power of the concept.

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. I think taking parts from Rotman’s An Introduction to Homological Algebra may work in a similar vein, albeit with much more cutting. Because there is quite a bit, and because students are assumed to be mature and to be getting used to reading math, not just going to lecture, one can easily assign boring lemmas and the like as exercises or reading but only truly easy ones as exercises!

It began to sell well as soon as the war was over, in at about – annually. Yes, we did know then that research mattered for tenure, but our joy in teaching was somehow connected with our respective research.

It provided a clear and enthusiastic emphasis on the then naclane modern and axiomatic view of algebra, as advocated by Emmy Noether, Emil Artin, van der Waerden, and Philip Hall. I’ve already commented extensively on most of the standard algebra texts at my blog-you can find the relevant post here: Vinberg seems similar but more intense, so I imagine it would work well, too.

If you want an older book written by a master expositor and mathematician, then I think Herstein works better than Mac Lane. But macpane want to teach them algebra even more. Birkhoff and Mac Lane also want to teach their burkhoff to prove things, of course.

It also introduces the student to modules, but it does not insist on working with modules instead of vector spaces whenever possible, which is probably good, because modules often serve to slightly confuse without adding anything more than a bit of generality. This is when students should come face-to-face with having to understand birkhodf, or else.

Also, I apologize for such a long response.

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

Chapter 6 introduces noncommutative algebra through its simplest and most fundamental concept: Byalgeebra new concepts inspired by it had begun to influence homology theory, operator alyebra, the theory of topological groups, and many other domains of mathematics. 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.

Our book, first published 50 years ago, was intended to present this exciting new view of algebra to American undergraduate and beginning graduate students. It has a lot of linear algebra, which is good, and it is not too hard, but it requires some work. Theoretically, less-prepared students and more-prepared students would not only survive but also get from this something substantial. The truth, though, is that undergraduates are fairly unlikely to read their textbook. Exercises of the latter type serve the important function of familiarizing the student with the construction of a formal proof.

The revised edition differs only in minor rearrangements and additions. 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.

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.

Chapters give an introduction to the theory of linear and polynomial equations in commutative rings. Yet, I also think a course should start with basic material. Modern algebra prospered mightily in the decadesfrom functional analysis to algebraic geometry – not to mention our own respective researches on lattices and on categories.

Again, eventually the students should be given this stuff to them.