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.)

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 excl

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. 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.