Topics in Algebra

This is a now-venerable introductory text in abstract algebra, last updated in 1975, and still suitable for an upper-level undergraduate course or perhaps a graduate course. The author states (p.vii), “I have aimed the book to be, both in content and degree of sophistication, about halfway between two great classics, A Survey of Modern Algebra, by Birkhoff and MacLane, and Modern Algebra, by Van der Waerden.” The word “Topics” in the title is too modest; the book is actually a concise but thorough introduction to groups, rings, fields, modules, and linear algebra, with a few genuine miscellaneous “topics” thrown in at the end.

 

One thing I especially like about this book is that it has lots of concrete examples. It’s easy in abstract algebra courses to overdo the abstract part and never consider any specific examples of the abstract theory. Herstein has plenty of specific examples, both in the body and in the exercises. In particular he makes good use of 2 × 2 matrices to manufacture all sorts of examples of groups with particular properties. Another good example is giving an example of a polynomial that is not solvable in radicals; he goes one step further and shows a class of polynomials of prime degree p such that the polynomial has the symmetric group \( S_{p} \) as its Galois group (and so is not solvable by radicals for \( p \geq 5 \)).

 

Here’s a brief overview of the chapters. Chapter 1 is background about set theory and number theory. Chapter 2 is about groups, mostly homomorphism and quotient groups leading up to Galois theory later on, but also a little bit about counting with groups, and a section on the Sylow theorems. It includes a good bit on permutation groups. Chapter 3 is about rings, mostly a study of the different kinds of rings such as unique factorization domains.  Chapter 4 (a short chapter) is about vector spaces and modules, again looking forward to fields and Galois theory, but also covering dual spaces and inner product spaces. Chapter 5 is about fields, and is mostly about Galois theory, although it also covers specialized topics such as the transcendence of \( \pi \), construction with ruler and compass, and solvability by radicals. Chapter 6 in on linear transformations; this is really about matrix theory and would usually be covered today in a separate course on linear algebra rather than as part of an abstract algebra course. This is a long chapter (95 pages). Chapter 7 is the “topics” chapter, and covers the theory of finite fields, Wedderburn’s theorem that a finite division ring is a field, Frobenius’s theorem characterizing division rings over the reals, and quaternions (with an application to representing integers as a sum of four squares).

 

The book, excellent though it is, when considered as a text suffers from the probably-fatal handicap of a ridiculous list price: over $200 for a fiftyyear-old print-on-demand paperback of 400 pages. (It would make a great Dover reprint!) In contrast, Dummit and Foote’s Abstract Algebra, another excellent and more modern and comprehensive text from the same publisher, is $139.95 for a 944-page hardcover.  

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Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences from 2010 through 2021. His personal website is allenstenger.com. His mathematical interests are number theory and classical analysis.