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Algebra (AMS Chelsea Publishing), by Saunders Mac Lane, Garrett Birkhoff
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This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance.
This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach--emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s--was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors.
- Sales Rank: #700524 in Books
- Published on: 1999-02-11
- Original language: English
- Number of items: 1
- Dimensions: 6.50" h x 9.25" w x 1.50" l, 2.25 pounds
- Binding: Hardcover
- 626 pages
Review
"Nearly every ten years there seems to arrive a new edition of this now classical book the review of which the reviewer hardly can improve. The main advantage of the authors had been the introduction of thoroughly categorical concepts into algebra." ---- Zentralblatt MATH
"The book is clearly written, beautifully organized, and has an excellent and wide-ranging supply of exercises ... contains ample material for a full-year course on modern algebra at the undergraduate level." ---- Mathematical Reviews
Most helpful customer reviews
0 of 1 people found the following review helpful.
Clear exposition. Also a very well-made book--so call me stupid.
By 3rd Day Believer
Herewith, the dumbest (preliminary, in the sense of I just got it in the mail today) review of MacLane and Birkhoff, 3rd Edition that you'll read. This book is very well made, with quality paper, and a buckram binding Smyth-sewn in signatures. Also from a more substantive POV: The exposition seems clear. More to come as I work through the text. I guess some reviewers have read all 600 pages and worked all the problems? Fibbers!
7 of 11 people found the following review helpful.
very solid introduction
By CuriousStudent
i used this text as my first algebra book. if you just began math and want to see a solid introduction to modern math this is it; self-contained and introduces many important ideas that are commonly used in modern math but typically not taught in undergraduate math courses.
158 of 160 people found the following review helpful.
Superb, if read with the right outlook
By Victor A. Vyssotsky
Birkhoff and MacLane collaborated for much of their careers, and their "A Survey of Modern Algebra", first published in 1941, was an easy-to-read, easy-to-teach-from, easy-to-learn-from early fruit of their collaboration. This jointly written book "Algebra", first published in 1967 and vastly improved in the 3rd Edition, can be far more difficult to tackle unless one goes at it with understanding of how to approach it. It mostly reflects MacLane's approach, rather than Birkhoff's, and MacLane was not only brilliant, but unusual among pure mathematicians, perhaps even idiosyncratic; he finally died at an advanced age a few months ago, and his passion for his field is reflected in the fact that he continued to advise graduate students well into his 90s, just as he had advised me (and criticized my thinking incessantly) as a graduate student more than 50 years before.
MacLane was far less interested in any particular topic in mathematics, although he was a master of many, than he was in how one should think about mathematics to understand it, do it on one's own, extend it, and most important of all, recognize when one had fully though through a problem and solved it, as contrasted to having merely produced a plausible discussion of it.
I know of no book on pure mathematics more worth reading than this one, but in contrast to some other reviewers who are probably clearer thinkers than I, I have to tackle it with great patience and care. The secret of grasping it without getting bogged down is to keep constantly in mind that MacLane filled in details without being much interested in them except as necessary completion of exposition. So, when you read it, do not concentrate on details; concentrate on overall structure of thought and exposition and then, later, come back to absorb details. That was how MacLane worked, and that was how he tried to teach his students to work. The key question always in his mind was: what formulation of axioms and structure is fruitful for attacking the topic at hand, and how can we use that formulation to create an inexorable train of thought leading to important results? This book, "Algebra" is very much a reflection of that way of thinking.
So, when you first read this book, skip freely over much of the development of particular topics. Instead, spend a great deal of time thinking about definitions, and about the precise way in which key theorems are stated. Spend time and effort exploring the question of why seemingly trivial variations of these would be less fruitful, or could even lead one into error. Skip from one part of the book to another, without getting bogged down in any one part. Ask yourself also why certain topics and certain cases are excluded. E.g. right at the beginning of the discussion of quadratic forms is a simple definition which begins: "If V is a finite dimensional vector space over a field F of characteristic not 2, ..." Pause right there and ponder over why fields of characteristic 2 are excluded from this definition; just skim the next ten pages without studying them. If you think hard enough to see why fields of characteristic 2 must be excluded from the discussion, the entire ensuing discussion of quadratic forms becomes crystal clear.
Once you have mastered the style in which the material is presented, you can quite easily come back and follow the details. And if you do that, I hope you will find this ook as rewarding as I have.
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