Customer Reviews

A SURVEY OF MODERN ALGEBRA.

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I have just started reading this book, and already I am
enthralled by the beauty and elegance of the authors'
exposition. Assuming nothing more than an acquaintance with
school algebra and a little geometry, they develop
the basic properties of central algebraic structures, including
rings, groups and fields. These are treated by reference to
familiar examples, such as the ring of integers and the
rational, real and complex fields. Everything that one learned
in school algebra is to be found here, though, as is to be
expected, each topic is treated at a rigorous, mathematically
sophisticated level. In the first two chapters, the properties
of the integers and rational numbers are gradually examined,
ultimately down to the definition of addition and multiplication
on the basis of Peano postulates. The authors then consider
polynomials, the real and complex numbers, vector spaces, linear
algebra and other topics.
The writing style is clear, concise and elegant, with each new
concept being carefully defined as it is introduced. The proofs
achieve a satisfying balance between detail and brevity. Indeed,
reading the proofs and completing the exercises would do much, I
am sure, to enhance a reader's mathematical facility.

If you are interested in acquiring a deeper understanding of
algebra, this book should serve as an excellent introduction.

This is a classic book on Algebra. There is much that I like about it. It is exactly what the name suggests--a survey course. It briefly introduces all sorts of topics, including rings, fields, groups, galois theory, vector spaces, lattices, boolean algebras, and much more. It is written at a fairly elementary level and it generally doesn't go into a great amount of depth in each subject. Interestingly, many (more modern) algebra texts omit a number of rather basic topics in this book. Also, many modern books separate "linear algebra" from "abstract algebra", whereas this book takes a more integrated approach.

I find it exceptionally clear and easy to read. Many of the subjects are made particularly easy; there is a strong concrete flavour to the text. The authors provide good motivation for the material.

I think this book would make excellent reading material for someone who is planning to study algebra. I did not pick it up until early in graduate school, and I wish I had had access to it earlier, when I was first studying ring and field theory. It is a fantastic reference for intermediate students, since it covers just about all the basics of algebra, and does so in a very understandable way. I think this book would make a fine textbook for an undergraduate course as well.

Modern algebra is an extraordinary topic and Birkhoff and MacLane do a superb job of exploring it. However, as is often the case with mathematical texts, the material can be somewhat dry.