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.
