[Book Cover]

Algebra: Abstract and Concrete, 1/e

Frederick Goodman, University of Iowa

Published August, 1997 by Prentice Hall Engineering/Science/Mathematics

Copyright 1998, 335 pp.
ISBN 0-13-283988-1

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This innovative and versatile text starts by counting symmetries and then quickly gets into group theory. Symmetry (and geometry) is a theme throughout. Author has website as well. =  PEDAGOGIC FEATURES: 
Teaches students to work things out for themselves.

  • Helps students learn the thought-processes, patience, and persistence necessary to do mathematics and requires them to participate and investigate, starting on the first page.
  • There is intentionally no solutions manual.
Presents phenomena before concepts where practical, e.g.:
  • Introduces geometric examples of symmetry groups and computes group multiplication tables and matrix representations before defining groups.
Uses linear algebra and complex numbers throughout, requiring students to review and extend their knowledge of linear algebra. Uses of linear algebra become more sophisticated as the text proceeds.
Highlights all theorems (propositions, lemmas, corollaries) and definitions in boxes.
Features an abundance of high-quality exercises.
  • Integrates exercises with the text, by providing specific references to exercises in the text discussions and in the proofs of many results.
Provides manipulable three-dimensional graphics and color versions of other graphics from the text via the Internet. =  CONTENT FEATURES: 
Uses a “groups first” organization, and emphasizes symmetry as a theme throughout.
Introduces important classes of groups directly after the definition and first elementary results — symmetric groups, cyclic groups, and dihedral groups. Uses these classes of groups as examples to guide the further discussion of basic group theory.
Explores geometric aspects of group theory.
  • Analyzes the symmetry groups of regular polyhedra, following the fundamentals of group theory.
  • Provides templates for producing cardboard models of the regular polyhedra, and offers access to manipulable three dimensional graphics (e.g., for the symmetry axes of regular polyhedra) via the Internet.
  • Treats isometries of Euclidean space, and the “wallpaper” groups, in the Topics part of the text.
Discusses group actions and applications to counting problems and group structure.
Contains a thorough introduction to ring theory — polynomial rings, ideals and homomorphisms, integral domains, Euclidean domains, and unique factorization.
Covers field theory and Galois theory in three chapters.
  • The introductory chapter on Galois theory (in the Basics part of the text) is uniquely concrete, containing a complete analysis of the Galois correspondence for cubic equations, followed by a statement of the general result, for complex polynomials.
  • The remaining two chapters on Galois theory (in the Topics part of the text) provide a general treatment of the Galois correspondence for separable finite dimensional extensions, and the theory of solvability.
Includes appendices on set theory, logic, mathematical induction, and complex numbers. =

= I. BASICS. gif/chlist.gif 1. Symmetry. gif/chlist.gif 2. A First Look at Groups. gif/chlist.gif 3. Basic Theory of Groups. gif/chlist.gif 4. Symmetries of Polyhedra. gif/chlist.gif 5. Actions of Groups. gif/chlist.gif 6. Finite Abelian Groups. gif/chlist.gif 7. Rings. gif/chlist.gif 8. Field Extensions — First Look.

= II. TOPICS. gif/chlist.gif 9. Field Extensions — Second Look. gif/chlist.gif 10. Solvability. gif/chlist.gif 11. Isometry Groups. gif/chlist.gif Appendix A. Almost Enough about Logic. gif/chlist.gif Appendix B. Almost Enough about Sets. gif/chlist.gif Appendix C. Induction. gif/chlist.gif Appendix D. Complex Numbers. gif/chlist.gif Appendix E. Models of Regular Polyhedra. gif/chlist.gif Appendix F. Suggestions for Further Study. gif/chlist.gif Index. =  …• 1998,  = 335 pp.,  = Cloth  =  (0-13-283988-1)  =  (28398-6)  = MM0604 = Course Guide Page 392 = =  OTHER TITLES OF INTEREST:  = Herstein, Abstract Algebra = Rotman, A First Course in Abstract Algebra = Shifrin, Abstract Algebra: A Geometric Introduction

Table of Contents

    1. Symmetry.
    2. A First Look at Groups.
    3. Basic Theory of Groups.
    4. Symmetries of Polyhedra.
    5. Actions of Groups.
    6. Finite Abelian Groups.
    7. Rings.
    8. Field Extensions — First Look.

    9. Field Extensions — Second Look.
    10. Solvability.
    11. Isometry Groups.
    Appendix A. Almost Enough about Logic.
    Appendix B. Almost Enough about Sets.
    Appendix C. Induction.
    Appendix D. Complex Numbers.
    Appendix E. Models of Regular Polyhedra.
    Appendix F. Suggestions for Further Study.


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