Algebra

Rating:
Summary: THE algebra book, period.
Review: 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.

Rating:
Summary: THE algebra book, period.
Review: 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.

Rating:
Summary: THE algebra book, period.
Review: 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.

Rating:
Summary: Relatively readable book based on category theory
Review: I won the Slovak translation of this book in a competition at the high school. It had many useful definitions and theorems in it and its rigorous, but readable approach seemed attractive to me and almost converted me into a rigorous mathematician, but obviously it turned out to be too abstract at the end.

However, the main problem was that several years later, because of particle-physics motivations, I tried to learn some things about the Lie groups, Lie algebras (e.g. exceptional Lie algebras), and their representations, and initially I assumed that such a big book had to contain something useful about these topics. It did not, I think.

This book is very good if you want to learn most of the rigorous math, partly based on the generalized abstract nonsense (also known as category theory), as it was known decades ago - the definitions and basic theorems about the basic algebraic structures. But you must be ready that many newest subjects - and those that are hot today - are not covered.

The book contains many problems - and these problems are solved in another book, which you may find very useful.

Rating:
Summary: bold and beautiful
Review: 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!

Rating:
Summary: A readable text using notation similar to Jacobsen.
Review: 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.