A Survey of Modern Algebra (Akp Classics)

Rating:
Summary: A beautiful text
Review: Fashions come and go, even in mathematics textbooks. Books from the 60s and 70s often have a set-theoretic flavor to them, because that was the fashion. Few books of any period are well written enough to resist such aging. spivak/calculus is one, and birkhoff&maclane is another.

This book strips algebra bare. I'm not talking about kiddie stuff like y=mx+c, I mean answers to the questions 'what does it mean to add two numbers together?' 'what is a real number'? Now there's no getting away from the use of sets and logic for these sorts of questions, but B&M do it with such elegance and clarity of exposition that it seems perfectly natural. And when you think about it, they're answering pretty fundamental questions; once that your school teachers glossed over. You can add 2 apples to 3 apples and count 5 apples, and maybe 2.5 and 3.5 apples make sense, but on what logical basis can you say that pi + pi is 6.28... given that you can never have exactly pi apples? Does saying you have a real number of anything make sense?

Rating:
Summary: A beautiful text
Review: Fashions come and go, even in mathematics textbooks. Books from the 60s and 70s often have a set-theoretic flavor to them, because that was the fashion. Few books of any period are well written enough to resist such aging. spivak/calculus is one, and birkhoff&maclane is another.

This book strips algebra bare. I'm not talking about kiddie stuff like y=mx+c, I mean answers to the questions 'what does it mean to add two numbers together?' 'what is a real number'? Now there's no getting away from the use of sets and logic for these sorts of questions, but B&M do it with such elegance and clarity of exposition that it seems perfectly natural. And when you think about it, they're answering pretty fundamental questions; once that your school teachers glossed over. You can add 2 apples to 3 apples and count 5 apples, and maybe 2.5 and 3.5 apples make sense, but on what logical basis can you say that pi + pi is 6.28... given that you can never have exactly pi apples? Does saying you have a real number of anything make sense?

Rating:
Summary: This is how algebra texts ought to be written
Review: 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.

Rating:
Summary: A smorgousborg of symmetries of the square
Review: 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.