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Integers, Polynomials, And Rings

Ronald S. Irving
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Ioana Mihaila
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Integers, Polynomials and Rings was developed, as the author explains in the preface, from a set of notes written for an “Introduction to Modern Algebra for Teachers” course. As such, the book is meant to be a structurally different abstract algebra textbook.

The most striking difference in the layout of the book is the absence of “end of section” exercises. Rather than that, the problems are spread throughout the text, and they play a role in supplying the necessary proofs for the results included in the text. Students using this book are expected to first discover mathematical ideas by considering carefully selected examples, and then prove general theorems, by following steps outlined in more exercises. The book is written for a two-quarter sequence, but it contains enough material to discuss in a year-long course. My estimate is that one semester would not be sufficient for covering enough of the contents to leave the reader with a good understanding of abstract algebra and its uses.

With an example-driven book, one may worry about the text’s ability to cover enough meaningful content. This is not the case at all. The author is quite skillful in guiding the student from the division theorem for integers, to Euler’s theorem and Fermat’s (little) theorem, to polynomial rings, and finally to more general Euclidian rings. Field extensions and a chapter on finite fields are included in the third part of the book. An unusually thorough discussion of quadratic, cubic, and quartic polynomials is included in chapter 10 (the author designates part of it as optional material).

Conspicuously missing from this treatment of abstract algebra is group theory. In its current form, the book is very unitary and it has a good flow. Introducing groups in it would probably be awkward, and, as the author points out, it wouldn’t do the theory of groups justice. For this reason, I would probably not use this book for a general “math major” abstract algebra course. In addition, I feel that a math major should have more exposure to abstraction, and in this book it is occasionally left to the instructor to emphasize that all the concepts presented are part of a general picture, to which we refer as “algebraic structures”.

Integers, Polynomials and Rings is a unique book, and should be extremely useful for an audience of future high school teachers. It would also be a valuable supplement for students taking a traditional abstract algebra course, especially since it is very readable.

Ioana Mihaila ( is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is interested in mathematics competitions.

Introduction: The McNugget Problem.- Induction and the Division Theorem.- The Euclidean Algorithm.- Congruences.- Prime Numbers.- Rings.- Euler's Theorem.- Binomial Coefficients.- Polynomials and Roots.- Polynomials with Real Coefficients.- Polynomials with Rational Coefficients.- Polynomial Rings.- Quadratic Polynomials.- Polynomial Congruence Rings.- Euclidean Rings.- The Ring of Gaussian Integers.- Finite Fields.- Index.