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Abstract Algebra: An Introduction to Groups, Rings and Fields

Clive Reis
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
, on

The book covers the same core material as all competing textbooks do, so the job of the reviewer is to discuss the few things in which the book does differ from its peers.

First, the set of topics covered is much smaller than usual. This is because of a very long introductory part that takes 100 pages and four chapters and contains material that is typically part of the prerequisites for an abstract algebra course. There is also a 60-page appendix. So only two-thirds of the book is devoted to abstract algebra, which certainly comes at a price. The theorems are about the same as in similar textbooks, but the exercises are significantly easier and the text connecting the results is terser. To the book’s credit, there are combinatorial applications that this reviewer has not seen in other abstract algebra textbooks, such as mutually orthogonal Latin squares or Burnside’s Lemma.

There is certainly a place for an algebra textbook on the easy side of the scale, with lots of introductory material, and simple exercises. The students of this reviewer often find the existing textbooks too difficult. However, these same students may well be reluctant to embrace this book as the reader-friendly alternative, since the exposition is terse and none of the exercises come with solutions, or even, hints, which make self-study difficult.

Miklós Bóna is Professor of Mathematics at the University of Florida.

  • Logic and Proofs
  • Set Theory
  • Cartesian Products, Relations, Maps and Binary Operations
  • The Integers
  • Groups
  • Further Properties of Groups
  • The Symmetric Groups
  • Rings, Integral Domains and Fields
  • Polymial Rings
  • Field Extensions
  • Latin Squares and Magic Squares
  • Group Actions, the Class Equation and the Sylow Theorems
  • Isometries
  • Polya–Burnside Enumeration
  • Group Codes