Customer Reviews

Algebra

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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.

After getting frustated by nearly all the so-called "authoritative" books on abstract algebra (Lang, Hungerford, Jacobson), I really can say that MacLane/Birkhoff is the best die-hard classic on algebra. Now I must stress that this book IS NOT out-of-print: the third edition is actually published by AMS/Chelsea.

There's an interesting thing about the evolution of this book: the first edition has become famous among mathematicians, because it brought for the first time an elementary exposition of categories and universal constructions, directly from the horse's mouth (MacLane founded the theory of categories together with S. Eilenberg; Birkhoff was the creator of the theory of lattices), which is used as a basic tool throughout the book; it also contained unusual topics such as multilinear algebra and affine and projective spaces, but no Galois theory. The second edition has gained a chapter on Galois theory, but has lost the part on affine and projective spaces.

The third edition is the best! It has recovered the part which was lost in the second edition, and had its exposition considerably polished. While most other books expose abstract algebra as a ugly, prawling monster, MacLane/Birkhoff manage to explain quite esoterical topics (many of them created and/or developed by themselves) in a surprisingly natural and tasty way (compare it with the dry, encyclopaedic style of Hungerford and Lang); although quite big, the book supports several ways of reading and teaching its parts without sacrificing clarity. Another great quality: it is INSPIRING, in the sense that it develops a powerful algebraic intuition, which is, in my opinion, the main obstacle one has to face to learn algebra.

It has several sections not present in most introductory texts -- affine and projective geometry, multilinear algebra, and linear algebra (the latter only seen in Herstein's Topics in Algebra), category theory, and lattice theory. The first few chapters use permutations a lot for examples, later it uses matrix groups. We are talking about the 3rd edition here -- don't get an earlier edition!

After getting frustated by nearly all the so-called "authoritative" books on abstract algebra (Lang, Hungerford, Jacobson), I really can say that MacLane/Birkhoff is the best die-hard classic on algebra (I've already reviewed an out-of-print edition here at Amazon, but since that review is not reproduced in this edition's page, I'm doing it myself).

There's an interesting thing about the evolution of this book: the first edition has become famous among mathematicians, because it brought for the first time an elementary exposition of categories and universal constructions, directly from the horse's mouth (MacLane founded the theory of categories together with S. Eilenberg; Birkhoff was the creator of the theory of lattices), which is used as a basic tool throughout the book; it also contained unusual topics such as multilinear algebra and affine and projective spaces, but no Galois theory. The second edition has gained a chapter on Galois theory, but has lost the part on affine and projective spaces.

The third edition is the best! It has recovered the part which was lost in the second edition, and had its exposition considerably polished. While most other books expose abstract algebra as a ugly, prawling monster, MacLane/Birkhoff manage to explain quite esoterical topics (many of them created and/or developed by themselves) in a surprisingly natural and tasty way (compare it with the dry, encyclopaedic style of Hungerford and Lang); although quite big, the book supports several ways of reading and teaching its parts without sacrificing clarity. Another great quality: it is INSPIRING, in the sense that it develops a powerful algebraic intuition, which is, in my opinion, the main obstacle one has to face to learn algebra.

Garrett Birkhoff and S. MacLane's _A Survey of Modern Algebra_ introduced U.S. undergraduates to the (axiomatic) algebra of Emmy Noether and Emil Artin, with elementary topics useful in applications in science and engineering. Birkhoff-MacLane has a place for algebraic number theory, but puts it in its place---Chapter 14! Birkhoff-MacLane features Birkhoff's interests in congruence relations (c.f., universal algebra), partially ordered sets (c.f., lattice theory), and linear algebra and geometry.

MacLane-Birkhoff's Algebra strives to teach algebra using the spirit and the ideas of category theory. Thus module theory is central to the text. However, this text is in theory accessible to undergraduate students, because the level of abstraction increases gradually, the examples are elementary, proofs are given in detail, and most problems can be solved easily (in the beginning chapters). These features make MacLane-Birkhoff a complement to Lang's Algebra, which uses category theory.

(Also, MacLane-Birkhoff does use ideas from lattice theory and universal algebra more than other texts and has a particularly detailed development of linear and multilinear algebra.)

For a more comprehensive graduate textbook, I would recommend Grillet's "Algebra" (which should replace Lang's book except in isolated populations of algebraic number theory).

I had the first edition of this book as a student about 30 years ago and found that although the text was for the most part readable, the notion of universal element notion from category theory was introduced in a baffling way very early on and this was the stumbling block.

I've been rereading mathematics books out of pleasure of late and wanted to read this one, so I looked up the Amazon reviews. It was written that you should get the third edition. I ordered it and was not disappointed. The stumbling block has been recognized by the authors and pushed to much further in the book. A new chapter on Galois Theory has been added.

The contents of this book is more basic in general than say Lang's book and the reader is not left to fill in the missing pieces and it is generally more readable. What's more the exercises are well chosen to consolidate the learning.

The book is not encyclopedic, but to my mind constitutes a very solid modern grounding for any mathematical student.

This is a fabulous book. I have had basically one year of effectively 'algebra for non-mathematicians' at a graduate level, and have a background in Physics and Engineering. I find this book to be the perfect next step in terms of understanding things a bit more deeply than Dummit&Foote, in particular because they bring in some wider issues and connect them to the basic material of group theory, ring theory, modules vector spaces etc. In other words it covers the same or similar material as Dummit & Foote, but adds in some wider examples (more lattice theory, operator theory) as well as a higher level of sophistication because it brings in a category theory perspective for example. It seems that this book is really oriented toward seeing the big mathematical picture implicit in lots of physics. So in that sense, for me, it's a fantastic mathematical physics book, without explicitly being so.

Let's start off with the bad: Not having a marker for the end of each proof annoys me. Some of the terminology is a little awkward. (For example, the word "homomorphism" appears once in the entire text, to note it as an alternative to their usage of the word "morphism".) There are a few minor typos here and there. Also...well, actually, that's it. I can think of nothing else negative to say about this book.

I make no claims to being a math prodigy; I pursue it simply because I enjoy it. When I read math, it's a struggle. What I found so remarkable about this book is that I was not struggling, despite the complexity of the material. That's not to say I found nothing in here difficult, because it's all difficult to some degree by the very nature of the material, but that the moments of frustration were few and far between. I will admit that I had familiarity with probably about 30% of the material when starting, so it is possible my opinion would be different had I not known what a group was beforehand. I kind of doubt that though, the exposition seemed nearly ideal. Beyond being a smooth read, there was no overstatement. Everything said added to the content.

The book covers a lot of territory in only 600 pages, but none of it should be a surprise to see in an algebra text: groups, rings, modules, Galois theory, a little on categories, etc. I think the chapter on matrices should be pointed out in particular. This is a better explanation of how matrices and their entries can and ought to be interpreted than any other I have read, including Axler! Admittedly, Axler does a slightly better job with some of the other linear algebra topics, such as characteristic polynomials.

Prerequisites for this book are extremely slim - naïve familiarity with the real and complex number systems are the only absolute necessity. However, anyone taking this on with nothing more than the bare minimum is going to be in for quite a challenge. You should already have some comfort with reading texts filled with theorems and proofs.

This is possibly the best textbook I have encountered to date. If you want to learn algebra, I can't imagine a better single place to look to get a well-rounded exposure to the subject.

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.

This text is a very readable presentation of first year graduate abstract algebra. The material is presented with notations similar to that of Jacobsen in his "Basic Algebra" texts, and is useful as a review text for qualifiers, or for independent study.