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A Concrete Introduction to Higher Algebra

Lindsay Childs
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Art Gittleman
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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.

BLL* — The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

1 Numbers 2 Induction 3 Euclid's Algorithm 4 Unique Factorization 5 Congruences 6 Congruence Classes 7 Applications of Congruences 8 Rings and Fields 9 Fermat's and Euler's Theorems 10 Applications of Fermat's and Euler's Theorems 11 On Groups 12 The Chinese Remainder Theorem 13 Matrices and Codes 14 Polynomials 15 Unique Factorization 16 The Fundamental Theorem of Algebra 17 Derivatives 18 Factoring in Q[x],I 19 The Binomial Theorem in Characteristic p 20 Congruences and the Chinese Remainder Theorem 21 Applications of the Chinese Remainder Theorem 22 Factoring in Fp[x] and in Z[x] 23 Primitive Roots 24 Cyclic Groups and Primitive Roots 25 Pseudoprimes 26 Roots of Unity in Z/mZ 27 Quadratic Residues 28 Congruence Classes Mopdulo a Polynomial 29 Some Applications of Finite Fields 30 Classifying Finite Fields