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Michael Artin
Prentice Hall
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Gizem Karaali
, on

Michael Artin’s Algebra is perhaps not yet considered a classic, but for many years it has been a serious contender for textbook of choice in the standard undergraduate abstract algebra course. So I was very excited to have the opportunity to read it critically for the MAA Reviews. After a few sessions of careful reading I can now happily say that my enthusiasm was completely justified.

This was certainly a good read. The book is well-written and the author’s style adds an appealing personal touch, though I still wish the non-English epigraphs were accompanied by their translations — at least the verbal ones. In terms of content, the book covers all the classical and standard material, but also introduces students to all the fun stuff that I have to pull out of my hat when I am using a more traditional text. Case in point: Symmetries of plane figures and crystallographic groups. This is such a beautiful topic, and perfectly appropriate for this level, but Herstein and Dummit & Foote do not touch it with a long pole. And of course Lang or Hungerford could never be expected to stoop so low…

Another case in point: Group representations. Artin justifies the inclusion of this topic in an undergraduate introductory text by saying “If chemists can do it, why can’t we?” I couldn’t agree more! I believe that it is our duty to introduce our students to what group theory means to their physicist and chemist friends, and the road to that goes through a path into representation theory. And for those who could not care less about representation theory, there is a chapter on quadratic number fields and one on Galois theory for alternative capstone experiences at the end of the term.

The book contains a lot more than what a typical one-semester algebra course could cover, and this is intentional: the author expects the instructor to pick and choose. This flexibility allows for many different kinds of courses to be taught from the same text. For instance, if your students take abstract algebra immediately after calculus and a mainly computational introduction to linear algebra, Artin’s book can be used to introduce abstract algebraic notions via the development of more advanced ideas in linear algebra, thus solidifying the linear algebra background of your students as they learn basic algebra. If on the other hand your students took a proof-based linear algebra course already (like the typical student at my own institution who comes into the abstract algebra classroom), then you can go very quickly through the linear algebra material that is covered in detail early on, and have the chance to touch upon more of the fun stuff later. It is highly likely that some of the linear algebra material will be new to any student, as Artin touches upon the Jordan canonical form, the spectral theorem, bilinear forms, matrix exponentials, and other such advanced topics. Even if these sound familiar to your students, a quick recap can never hurt.

Teaching a course following this book could be a joy. My minor annoyance for the term used for group actions (Artin calls them group operations) is quite easily overshadowed by my overwhelming support for the early and often use of linear groups as examples. Unlike more traditional texts Artin does not solely depend on permutation groups as the major examples of groups; he develops a substantial amount of theory for linear groups as well (including a brief introduction to Lie algebras). There is some amount of topology, geometry and analysis that is referred to in various instances, which is not typical, but I think such references could help our students notice earlier than they usually do that in fact these branches of mathematics are not really unrelated, and that they actually benefit from connections and interactions. Looking carefully, it is clear that the amount of topology, analysis and geometry required is not substantial and allows Artin to cover much more significant mathematics. Therefore I would be more than willing to give him a break for writing a book which is technically not completely self-contained.

To many, Artin’s book may have seemed like an interesting experiment in the first edition. In this new edition the book has matured, and is, I believe, ready to compete against anybody’s personal favorite. Take a look!

Gizem Karaali is assistant professor of mathematics at Pomona College and an editor of the Journal of Humanistic Mathematics. As a representation theorist by training, she is obviously quite partial to introducing the subject in the undergraduate curriculum.

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