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A Concrete Introduction to Higher Algebra

Lindsay N. Childs
Publication Date: 
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Undergraduate Texts in Mathematics
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
P. N. Ruane
, on

This edition of Lindsay Childs’ book is 85 pages longer than the previous version, which was published in 1995. Since then, many sections have been rewritten and some sections from the second edition have been omitted.

Additional material is said to include further coverage of quotient groups, with applications to quadratic reciprocity and the cardinality of finite fields. There is also a new application of Eisenstein’s irreducibility criterion to Chebyshev polynomials and a new proof of Rabin’s theorem. Other additions include Luhn’s formula, Montgomery multiplication and Blum-Blum-Shub pseudorandom numbers.

The stated prerequisite knowledge consists of pre-calculus algebra, first year calculus and some prior knowledge of linear algebra, and the author’s general aim is to introduce rings, fields, groups and homomorphisms by means of ‘concrete’ examples from the integers and polynomials. There is a stated emphasis on congruence classes to motivate the introduction of finite groups and finite fields.

Structurally, the book consists of 27 chapters, organised into seven parts with the headings Number, Congruence Classes and Rings, Congruence and Groups, Polynomials, Primitive roots, Cyclic Groups and Cryptography, Finite Fields and Factoring Polynomials. However, given the large amount of material contained within 600 pages, the author recommends five truncated pathways through the book, and each pathway is the suggested basis taught course, as follows:

  1. Abstract algebra up to finite fields
  2. Classical algebra.
  3. Number theory.
  4. Applicable algebra
  5. A shortened course with emphasis on polynomials.

So far, so good, but what are the distinguishing features of this publication? Well, in keeping with the intended emphasis on congruence, the flavour is mainly number-theoretic (as suggested by 3 above). Consequently, the vast majority of examples, and the general context in which ideas are introduced, are expressed in terms of the integers, rational numbers, modular arithmetic and polynomials

In fact, the reader has to wait until page 248 before examples of algebraic ideas appear in wider range of contexts, such as the symmetry group S3 and the dihedral group D3. And, apart from brief mention of vector product and matrices (as zero divisors), the range of examples, although large in number, tend to serve the needs of number theory. In particular, there is little or no mention of permutation groups, quaternions or groups of functions, and only the above brief reference to matrices and geometric transformations (nothing on frieze patterns etc). But such comments really reflect the mismatch between the actual contents of the book and the expectations generated by its title.

Nonetheless, I fully agree with the publisher’s claim about this book being a ‘readable and informal introduction to higher algebra’. This is achieved by the clarity of the exposition and the provision of the large number of exercises and examples throughout the book. I specifically liked the introduction to finite fields and the eventual application to Latin squares and BCH codes.

Of course, because this book can serve as both an introduction to number theory and abstract algebra, sacrifices have to be made with respect to its algebraic content. For instance, the treatment of groups is briefer than most, which is reflected by the above comments, and also by the lack of coverage of familiar topics such as the Sylow theorems, simple groups and the Jordan-Hölder theorem etc. However, the pay-off for such omissions is the wealth of less well-known ideas, such as Van der Waerden’s Theorem, Berlekamp’s factoring algorithm and Blum-Goldwater cryptography.

Finally, with regard to the re-writing of various sections of the book, it seems that this may have not included the early chapters, where I found several anomalies of expression. For example, the division algorithm is incorrectly stated on p. 27, and the notions of equivalence and equality are too readily conflated in the discussion of rational numbers (p. 5). Also, the introduction to equivalence classes and induced binary operations is somewhat brief, even though such ideas are reinforced by subsequent applications.

Apart from this, and despite one or two later misprints, the book has been written with a high degree of rigor and accuracy and I definitely recommend it for consideration as the basis of an alternative route into abstract algebra and its applications.

In his days as an undergraduate, and prior to taking courses on higher algebra, Peter Ruane benefited from a foundation course on logic, sets, relations and functions.


Preface.- Numbers.- Induction.- Euclid's Algorithm.- Unique Factorization.- Congruence.- Congruence Classes.- Rings and Fields.- Matrices and Codes.- Fermat's and Euler's Theorems.- Applications of Fermat's and Euler's Theorems.- Groups.- The Chinese Remainder Theorem.- Polynomials.- Unique Factorization.- The Fundamental Theorem of Algebra.- Polynomials in Q[x].- Congruences and the CRT.- Fast Polynomial Multiplication.- Cyclic Groups and Cryptography.- Carmichael Numbers..- Quadratic Reciprocity.- Quadratic Applications.- Congruence Classes Modulo a Polynomial.- Homomorphism and Finite Fields.- BCH Codes.- Factoring in Z[x].- Irreducible Polynomials.- Answers and Hints to the Exercises.- References.- Index.-


akirak's picture

The transition to abstract mathematics is difficult for most students whose aspirations do not include becoming research mathematicians. The author's approach is to build on students' background in high-school algebra and calculus in a sophomore-junior course. His goal is to reach the classification of finite fields. Each section has ample exercises, some computational, and many building proof skills, which are appropriate to the course level.

The 600 pages contain much more material than can be covered in a semester course while the varied topics allow choices of emphasis for courses with differing objectives. The author outlines five possible courses: Introduction to Abstract Algebra, Classical Algebra, Number Theory, Applicable Algebra, and a course emphasizing polynomials. This text would work well for any of these courses. There are many applications to factoring, cryptography, and a whole chapter on the fast Fourier transform and fast polynomial multiplication for those wishing an applied emphasis.

The 27 chapters are partitioned into seven sections; Numbers, Congruence Classes and Rings, Congruences and Groups, Polynomials, Primitive Roots, Finite Fields, and Factoring Polynomials. Part 1, containing five chapters, looks more like the start of a number theory text but follows the author's approach in building on students' concrete experience. Institutions with a separate number theory course could start at a more advanced level if this material were a prerequisite.

The author's Classical Algebra course includes most of the first four parts of the text excluding some of the applications. Such a course is ideal for most typical Mathematics majors, many of whom hope to become secondary teachers. It makes a graceful transition to abstract mathematics and gives students a resource with much additional interesting material for them to ponder. The author carefully notes the changes from the second edition which include extensive revision and about 85 pages of expanded material.


Art Gittleman ( is Professor of Computer Science at California State University Long Beach.