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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This book also does not match a clear unorthodox vision of how the subject should be taught, unlike, say, Aluffi or Adkins. I think too much about algebra pedagogy and textbooks. One of us would draft a chapter and the other would revise it.
Terminology and notation maclan has become outmoded since the Revised Edition was published in have been brought up-to-date; material on Boolean algebra and lattices has algbera 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.
The authors are quick to indicate applications and careful to motivate and illustrate abstractions.
The lecture ought to focus on the main ideas, examples, and results, and the book ought to clearly show what is important, very important, 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. Modern algebra prospered mightily in the decadesfrom functional analysis to algebraic geometry – not to mention our own respective researches on lattices and on categories.
Supplementing this with another book to get some coverage birlhoff representation theory and homological algebra is probably ideal.
A survey of modern algebra / by Garrett Birkhoff and Saunders MacLane – Details – Trove
Similarly, books like Cohn, Grillet, and Jacobson can be too advanced or too focused on being references. In we prepared a carefully polished second edition, in which polynomials over general fields were treated before specializing to the real field.
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. Those desiring a text replete with possibilities for courses tailored to various kinds of students should welcome this new edition.
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Linear algebra is first, because students have the best intuition there; ring theory is next, because the examples and bigkhoff are nicer there than in groups and the quotient construction is easier. 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.
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. For the social scientist whose mathematical studies have reached through the calculus, this book can confidently be urged as the thing to study next.
I often found it dry, too. Only the last five chapters remain unchanged. The rejuvenation of algebra by the systematic use of maclqne postulational method and the ideas and point of view of abstract group theory has been one of the crowning achievements of twentieth century mathematics. Indeed, I find his writing quite agreeable. 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.
Macoane over twenty years this text has been the “classic” work in its field. It is a great supplementary text, however. I doubt they’ll be able to fully appreciate the enormous generalization and unification it provides without a giant stock of examples which is why I love Emily Riehl’s Category Theory In Context as the definitive introduction to category birkhofc for mathematics students.
That may overstate things, since my friend probably knows more mathematicians than students who got fed up and left. Some combination of the approaches of the above books ought to make an excellent undergraduate course. Only in this way will they be able to appreciate the full richness of the subject. You should have never given up on the project. Developing the ability to take some vague notion birkkhoff work out the details is key to graduate education.
Nowhere can teachers better catch today’s spirit of mathematics. Birkhoff and Mac Lane’s book first appeared inand van der Waerden’s book first appeared in The authors express the belief that “for many students, the value of algebra lies in its applications to other fields: Modern algebra also enables one to reinterpret the results of classical algebra, giving them far greater unity and generality.
On the other hand, calculus has been pushed into high school, and introductory courses are substantially easier and less rigorous than in the days of Apostol, Courant, and Maclne. Yes, we did know then that research mattered for tenure, but our joy in teaching was somehow connected with our respective research.
Usually,for strong undergraduate courses, he’d supplement Algebra with examples drawn from he and Birkoff’s classic undergraduate text, A Survey Of Modern Algebra.
This book is distinguished also by the great clarity with which all details have been presented. There are today major directions that either did bkrkhoff exist over 60 years ago, or that were not then recognized to be so important. Getting across the rich motivation, ideas, analogies, history, and so forth should be an explicit goal, but it will not always be possible.