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Elements of Abstract Algebra

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This is a concrete introduction to abstract algebra, focusing on number theory and solvability by radicals. It is not a survey course, but it covers selected important portions of the standard topics of set theory, group theory, ring theory, and field theory. It emphasizes normal subgroups, permutation groups, and composition series in group theory, polynomial rings and ideals in ring theory, and algebraic numbers and Galois theory of field theory. Its concision (only 200 pages) comes from its selectivity and from having only a few worked examples.

This is a 1984 corrected reprint of a 1971 text. The book is modular: it is organized as a series of short (1–3 page) articles, with each article explaining some idea and giving a handful of exercises on that idea. The exercises are quite difficult; a few deal with concrete or numerical examples, but most ask for proofs and they often introduce additional concepts that are not used in the main narrative.

The applications treated include solvability of polynomial equations by radicals (in particular there is an example of a quintic that is not solvable), impossibility of the classical Greek construction problems (trisecting an angle, doubling the cube, and squaring the circle), and constructibility of regular polygons (including an explicit construction of a regular 5-gon and the numerical work to construct a regular 17-gon). Algebraic numbers are used to solve some classical number theory problems, including representation of integers as a sum of two squares and Fermat’s Last Theorem for exponent 3. The ideal theory section is the only one that doesn’t seem closely tied to particular problems; it is motivated by Fermat’s Last Theorem and the failure of unique factorization in number fields, but having re-established unique factorization through ideals, we don’t do anything with it.

Although it is oriented towards several classical problems, the motivation often feels weak because the exposition is organized by subject area and not by problem. For example, composition series, the Jordan–Hölder Theorem, and solvable groups are defined and their properties derived on pp. 53–57, but we don’t get around to using them until the Fundamental Theorem of Galois Theory on p. 112.

This book is not very similar to any other on the market. Another bargain-priced Dover book that is more accessible (due to having more and easier exercises), but without much broader coverage, is Pinter’s A Book of Abstract Algebra. The classical survey work here is Birkhoff & Mac Lane’s A Survey of Modern Algebra, but that is a genuine survey book and much more comprehensive than the present book. The present book has enough coverage for an undergraduate course, but will probably be too difficult for most undergraduate students because of the lack of examples and of easy exercises. It might be useful for an honors course, or for a course that emphasizes inquiry learning. I’ve found it very useful as a reference, because it is easy to find things in it and it is clearly written and well cross-referenced.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

Date Received: 
Friday, December 31, 2010
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Allan Clark
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Allen Stenger
  • Foreword
  • Introduction
  • I. Set Theory
    • The notation and terminology of set theory
    • Mappings
    • Equivalence relations
    • Properties of natural numbers
  • II. Group Theory
    • Definition of group structure
    • Examples of group structure
    • Subgroups and cosets
    • Conjugacy, normal subgroups, and quotient groups
    • The Sylow theorems
    • Group homomorphism and isomorphism
    • Normal and composition series
    • The Symmetric groups
  • III. Field Theory
    • Definition and examples of field structure
    • Vector spaces, bases, and dimension
    • Extension fields
    • Polynomials
    • Algebraic extensions
    • Constructions with straightedge and compass
  • IV. Galois Theory
    • Automorphisms
    • Galois extensions
    • Solvability of equations by radicals
  • V. Ring Theory
    • Definition and examples of ring structure
    • Ideals
    • Unique factorization
  • VI. Classical Ideal Theory
    • Fields of fractions
    • Dedekind domains
    • Integral extensions
    • Algebraic integers
  • Bibliography
  • Index
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Friday, December 31, 2010