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Essentials of Modern Algebra

Cheryl Chute Miller
Mercury Learning & Information
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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See Mark Hunacek’s review of the first edition. The author’s preface to this second edition indicates two changes:

  • Chapters 1–3 have been reorganized to make the material less “tightly packed” than before. The group of units in \(\mathbb{Z}/n\mathbb{Z}\) is introduced earlier in order to provide more examples of groups, and homomorphisms are postponed to chapter two.
  • Twelve biographical profiles of mathematicians have been added, one at the end of each chapter. Rather than sticking to the usual suspects, the author says she “decided to include information about some who are not as commonly heard about,” focusing on mathematicians “who had to overcome struggles due to race, gender, religion, age, or sometimes even health to persevere.

The weird definition of \(a\pmod{n}\) used in the first edition is retained even though chapter 0 includes a discussion of equivalence relations. As Hunacek notes in his review, this definition should lead to writing things like \(5\!\pmod{4}=1\) rather than \(5\equiv 1\pmod{4}\).

The twelve biographical essays are short accounts in the style of a CV: birth, education, degrees, academic positions, death, honors. Most give no information about the subject’s mathematical work. There are a few minor errors. Given the choice to focus on overcoming struggles, there is often a discussion of when someone’s work was “accepted” or “recognized,” but these vague terms are not usually clarified. For example, it is not clear to me what this means: “Sadly, only in 2001 did the mathematics community officially recognized Haynes as the first African American woman to earn a PhD in mathematics.” (p. 242, biography of Euphremia Lofton Haynes)

As Hunacek’s review says, this is a usable but not exceptional textbook. The exercises at the end of chapters are mostly easy, but the projects enhance them in significant ways. The inclusion of Galois theory (restricted to characteristic zero or finite base fields) is a very good feature.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He has taught abstract algebra more times than he cares to count.

0. Preliminaries
1. Groups
2. Subgroups and Homomorphisms
3. Quotient Groups
4. Rings
5. Quotient Rings
6. Domains
7. Polynomial Rings
8. Factorization of Polynomials
9. Extension Fields
10. Galois Theory
11. Solvability
Hints for Selected Exercises