Let's begin with the conclusion: this is a truly wonderful book, one that anyone teaching abstract algebra should read and which should be pointed out to talented students, particularly those who want to know a little more about what and why abstract algebra is.
This book is volume 1 in the Algebra section of the Springer Encyclopaedia of Mathematical Sciences (and volume 11 in the overall series). The Encyclopaedia series started as a joint effort with the Soviet publisher VINITI, but has now evolved into a much more ambitious project, with new subseries covering many more areas. Some algebraic topics, such as Representation Theory, Number Theory, Algebraic Geometry, and Lie Groups and Algebras, have their own subseries, so the Algebra series does not cover those topics.
As the first volume in the Algebra series, Basic Notions of Algebra tries to give the "big picture": what the subject is about, what it is for, some hints of its history. Treatments of specific sub-areas are in other volumes of the Encyclopaedia series. The volumes in the original algebra series (to the extent I was able to determine) are:
Basic Notions of Algebra
Non-Commutative Rings, Identities
Infinite Groups and Linear Groups
Combinatorial and Asymptotic Methods of Algebra and Nonassociative Structures
Combinatorial Group Theory and Applications to Geometry
Representations of Finite-Dimensional Algebras
Finite Groups of Lie Type and Finite-Dimensional Division Algebras
As one can see, this is an ambitious series that attempts to provide an overview of many parts of contemporary algebra. Shafarevich's volume provides a fitting introduction to all this.
When trying to explain to people what algebra is, one can imagine two different strategies. The first is to attempt to define the subject, say as the study of "structures" or something along those lines. The second is simply to give a tour, pointing out the most important sights and their significance. Shafarevitch says a few words about algebra in general, arguing that it provides various structures that "coordinatize," in one way or another, various mathematical (and physical!) phenomena. He cites projective planes and quantum mechanics as two particularly rich examples of this.
On the whole, however, Shafarevich opts for the latter strategy. Starting with fields, he discusses many of the standard algebraic structures, explains their origins, the main theorems about each, and why they are interesting. In each case, he highlights connections with other parts of mathematics.
As a result, the book becomes a long argument by example. In effect, it displays how algebra can be deployed as "a way to understand mathematics," to use Sauders Mac Lane's famous description. So fields are defined in the usual way, but after giving the standard examples Shafarevitch mentions fields of rational functions, the function field of an algebraic curve, and fields of Laurent series. Groups are discussed first as symmetry groups, then as abstract groups, and the definition is followed by a description of the group of extensions of a module and of the Brauer group of a field. Crystallographic groups come up frequently and are treated in detail.
In the treatment of each kind of algebraic structure, Shafarevitch emphasizes the most important examples. When a simple short proof of a theorem is available, he gives it, but big theorems are simply cited without proof. Since many connections with other parts of mathematics are made, readers should expect to meet, here and there, notions that are new to them. That is usually not an obstacle: Shafarevich is very clear and easy to understand, and the dependence on other subjects is usually "local." So if one can't follow the explanation of how the notion of a Lie algebra is related to the invariant vector fields on a Lie group, one can just take it for granted and move on.
Most mathematicians learn something from this book. The examples are particularly well chosen, simple enough to understand and complicated enough to display the main features of the theory. Since Shafarevich is writing a kind of survey, he highlights what is important and ignores technical details.
It's all too easy, when teaching or learning algebra, to lose sight of the forest. I can't imagine a better way to reorient oneself than reading through this book. Shafarevich provides reliable and interesting guidance, and he often has something new or unexpected to show us.
Shafarevich's writing is spare and elegant. Only occasionally do we see a flash of wit or the author's personality. At one point, for example, he discusses the application of the theory of representations of Lie groups in quantum mechanics. After a very clear explanation of the connection, he ends by noting that "Similar ideas have been widely developed over the last twenty years, finding applications also outside the domain of strong interactions. But at this point the author's scant information on these matters breaks off."
Miles Reid's translation is, in general, quite good, showing little sign of "translation English." One of the few instances in which something goes wrong is when he uses "octavions" to refer to Cayley's "octonions." I hope I'm right that this isn't standard usage, because I find it very ugly.
The book concludes with a useful, if uncompromising, bibliography. Reid notes that he asked Shafarevich whether "references to classics of the subject might seem old-fashioned to modern students. Shafarevich's reply was that just because we know of other people's bad habits, it doesn't follow that we should encourage them, does it?"
This is a wonderful book, one that will enrich your understanding of algebra and deepen your knowledge of mathematics as a whole. Tolle, lege!
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College, and the editor of MAA Reviews. He has taught abstract algebra more times than he can count, and he hopes he will be able to pass on to his students some of what he has learned from this book.