Algebra: Abstract and Concrete
Frederick M. Goodman
This text is an introduction to modern algebra for undergraduate
students, published by Prentice Hall in August, 1997. The book addresses
the expected topics -- groups, rings, and fields -- with symmetry as a unifying
theme. Presenting these topics is no doubt important, as the subject matter
is central and ubiquitous in modern mathematics.
However, the more important goal of this book is to introduce
students to the active practice of mathematics and to draw them away from
the view of mathematics as a system of rules and procedures. Students are
asked to participate and investigate, starting on the first page.
The first part of this text is suitable for beginners.
Any course on abstract algebra for beginners faces an enormous pedagogical
challenge: to bring students who have been raised on procedural mathematics
to start thinking about mathematics like mathematicians. It has to be expected
that students will have great difficulty in making this transition, just
as a person untrained in art has great difficulty to begin to see and draw
like an artist. The author is convinced that the apprentice artist had better
learn to draw by drawing, and the apprentice mathematician had better learn
to do mathematics by starting, immediately, to do mathematics. This thought
guides especially the crucial introductory sections of the text, which consist
of a concrete exploration of the symmetries of simple geometric figures.
The concreteness of this investigation is a great challenge to students,
who have learned to prefer an abstract and formal procedure to a concrete
- The text has better than average success in putting its
goals into practice. Phenomena precede concepts, to the extent that this
is practical. Examples are plentiful. Exercises, which ought to be the
heart of the course, are of high quality.
- The exposition is exceptionally clear and forceful.
- The text has a "groups first" organization.
In the first chapter, geometric examples of symmetry groups are introduced
and group multiplication tables and matrix representations are computed
before the definition of groups is given.
- Important classes of groups are introduced directly after
the definition and first elementary results: symmetric groups, cyclic groups,
and dihedral groups. These examples are then used to illustrate further
concepts of group theory.
- The symmetry groups of regular polyhedra are analyzed
in Chapter 4, following the fundamentals of group theory in Chapter 3.
Templates are provided for constructing cardboard models of the regular
polyhedra. Students like working with physical models of the polyhedra
in connection with discussing their geometry and symmetry.
- A thorough, but concise, treatment of basic ring theory
contains the expected material on polynomial rings, ideals and homomorphisms,
integral domains, Euclidean domains and unique factorization.
- The introductory treatment of Galois theory is uniquely
concrete, containing a complete analysis of the Galois correspondence for
cubic equations, followed by a statement of the general result, for polynomials
over the rational numbers.
- Linear algebra and complex numbers are used throughout
the text. Most students will probably need to review this material as they
encounter it in the course; this is a virtue of the text. Uses of linear
algebra become more sophisticated as the course proceeds.
- Appendices on set theory, logic, mathematical induction,
and complex numbers are provided. Equivalence relations are treated in
the text proper, in connection with cosets in groups.
- There is no student solutions manual. The purpose of
the course is to teach students to work things out for themselves. A solutions
manual makes this nearly impossible, and only interferes with the instructor's
task of helping students to analyze problems.
- Inclusion of more advanced material on Galois theory
and isometry groups makes the text usable for a variety of courses and
with students of different backgrounds and levels of sophistication.
- Excellent graphics. Here are some of the graphics from
the text converted to vrml
format, for 3-D viewing over the web.