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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They do not avoid using universal properties, and they do not always bother to give students something concrete to hold on.
His paper, “On the Structure of Abstract Algebras” founded a new branch of mathematics, universal algebra. We have done this by illustrating each new term by as many familiar examples as possible.
This exposition of the elements of modern algebra has been planned with great skill, and the plan has been carried through very successfully. Modern algebra alegbra mightily in the decadesfrom functional analysis to algebraic geometry – not to mention our own respective researches on lattices and on categories.
Our book, first published 50 years ago, was intended to present this exciting new view of algebra to American undergraduate and beginning graduate students.
Birkhoff and Mac Lane also want to teach their students to prove things, of course. Chapter 6 introduces noncommutative algebra through its simplest and most fundamental concept: 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. The truth, though, is that undergraduates are fairly unlikely to read their textbook.
Developing the ability to take some vague notion and work out the details is key to graduate education. There are today major directions that either did not exist over 60 years algebrs, or that were not then recognized to be so important. That was characteristic of his freshness, his initiative, and his lack of respect for conformity; but it came as a slight shock to me at the time. In this way, I do not expect today’s students are substantially better than past students, birkhorf.
Usually,for strong undergraduate courses, he’d supplement Algebra alfebra examples drawn from he and Birkoff’s classic undergraduate text, A Survey Of Algbera Algebra. This book is distinguished also by the great clarity with which all details have been presented. We have also tried not to lose sight of the fact that, for many students, the value of algebra lies in its applications to other fields: In fact, they may be better when it comes to algebra.
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.
Modern algebra also enables one to reinterpret the results of classical algebra, wlgebra them far greater unity and generality. Those are perhaps worth substantial time, even if they are somewhat too odd for most courses. I’m going to soon review the 3rd edition of Saunders MacLane And Garrett Birkoff’s Algebra at my blog soon and this is the first time I’m really carefully reading it.
I think using many texts is often and in this case almost certainly a good idea. 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. I mentioned flaws of a few books in my answer.
Hungerford is a common alternative I did not discuss, but I have never found it very exciting. These ideas are still most relevant and worthy of enthusiastic presentation.
I am joking, if it is not clear. Surely, people appreciate the extant ones e.
Algebra – Saunders Mac Lane, Garrett Birkhoff – Google Books
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. On the other hand, calculus has ,aclane pushed into high school, and introductory courses are substantially easier and less rigorous than in the days of Apostol, Courant, and Spivak.
Moreover, many of the birkhogf in this text algbra help teachers to communicate this spirit to their students. The next year Mac Lane put group theory first, and set theory Boolean algebra last! While it can be used as a reference, it should rather be read through carefully over a period of years – one must think in terms of years if one wishes to absorb fully the material and to do the problems.
The group concept is applied systematically in Chapterson vector spaces and matrices. There are still many overqualified students and many underqualified students, and I can provide a constructive proof of that fact. This is a text on modern algebra that is particularly suited for a first year graduate course or for an advanced undergraduate course.
It is also worth noting Mac Lane does cover multilinear algebra pretty well, which is frequently forgotten. I think the needs of the graduate student are more complex, in a sense, than the undergraduate’s needs, because many are less obvious than in the undergraduate case.
Suggestions I am less sure what makes a really excellent graduate course in terms of extant texts. Such students are strongly advised to do supplementary reading, or at least browsing, in the references listed at the end.
Logic suggests the standard groups, rings, fields, modules, vector spaces, etc. Getting across the rich motivation, ideas, analogies, history, and so forth should be an explicit goal, but it will not always be possible.