Algebra: Abstract and Concrete

Frederick M. Goodman

Pedagogical Perspective:

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 problem!

Features:

• 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.