Galois Theory

We should definitely be teaching Galois Theory to undergraduates. We do, after all, force them to learn about groups, rings, and fields; we owe them a chance to see what those ideas can do. The problem of solving polynomial equations was what led to the birth of those algebraic ideas in the first place and so provides the perfect opportunity to put them to use. On top of it all, the Galois correspondence is a beautiful and satisfying result.

It’s crazy to teach Galois Theory to undergraduates. Most of them don’t even know the quadratic formula and don’t care about solving polynomial equations by radicals; numerical solutions are sufficient in most cases anyway. An honest account of the theory involves too many subtle notions such as normality and separability. The lead-up to the main theorems is too long, and the usual pay-off is an impossibility result. What value is there in knowing that something cannot be done? Much better to develop a theory with real applications and positive results.

I don’t know which of those arguments is closer to the truth. I did learn Galois theory as an undergraduate. I enjoyed it, but I also found it very difficult. One of the books I used to try to make some sense of it all was the first edition (1973) of Ian Stewart’s Galois Theory. It must have been one of the first books on the subject aimed at undergraduates. Fifty years later, here is a fifth edition of that book.

When writing on Galois Theory one must first decide what the subject actually is. At one extreme, one can assume students have learned in their abstract algebra course enough about groups, fields, and polynomials to allow us to jump right in, resulting in a short book. At the other extreme, one can assume only a minimal amount of algebra and develop the rest, making for a longer book in which there is much opportunity to develop related topics. Stewart chooses the latter, taking as known only linear algebra and the basics of the theory of finite groups. (So, for example, when he needs Cauchy’s theorem about groups with order divisible by a prime, he gives a proof.) Stewart also chooses to include several subjects that are not part of Galois Theory proper: the classical solutions of cubic and quartic equations, the “Fundamental Theorem of Algebra,” ruler and compass constructions (not just the three standard problems, but also constructing regular polygons), the Abel-Ruffini proof of the unsolvability of the general quintic, the transcendence of \(e\) and \(\pi\), parts of Gauss’s cyclotomic theory. He also spends a considerable time discussing the history of all this material. This is a meaty book that covers more than its page count suggests.

Stewart opens with a “Historical Introduction” that seems largely unchanged since the first edition. In any case, it displays no awareness of the work of more recent historians. (The section of the bibliography devoted to “Historical Material” includes only one reference from the last 20 years: Newmann’s 2011 edition of Galois’s mathematical writings.) The discussion of Babylonian “algebra” does not refer to Jens Høyrup’s work. Wilbur Knorr is not mentioned in the section on Greek construction problems. The work of Caroline Erhardt problematizing the iconic figure of Galois is not mentioned. Historical ideas and interpretations are not like mathematical theorems; they change over time. This chapter, as it stands, is disappointing.

Once past the historical introduction, this edition is very different from the first and second editions I grew up on. The major change seems to have come in the third edition. The first two editions dealt with general fields and field extensions from the beginning. “The fashion of the times,” says Stewart (p. xvi), “favoured generality and abstraction.” Beginning in the third edition, Stewart decided to adopt a “more concrete” approach. Instead of general fields, he initially focuses only on subfields of the complex numbers. The first 15 chapters work in this setting, from classical algebra to the Galois correspondence and the question of solution by radicals. Only in chapter 16 do we get “Abstract Rings and Fields” and begin to develop the general theory.

The rearrangement causes some problems. One advantage of working inside \(\mathbb C\) is the Fundamental Theorem of Algebra: we do not need to give an abstract proof of the existence of a field where our polynomial has a root or of a splitting field, since the roots are all complex numbers. Stewart decided that to do this honestly he should give a proof of the FTA, however. The result is that chapter two is one of the hardest in the book. The proof requires “a few ideas from elementary point-set topology and estimates of the kind we encounter early on in any course on real analysis” (p. 24). Perhaps this reflects the fact that British students tend to learn real analysis earlier than American students. Many of the students in my Abstract Algebra classes would have trouble following this proof.

(Stewart actually gives another proof of the FTA in chapter 23, one that uses Galois theory. It is much easier to understand. A proof of the existence of an algebraic closure of a general field is given in chapter 17.)

Another issue created by the rearrangement has to do with the “general equation of degree \(n\),” introduced in chapter 8 in order to prove that it has no solution by radicals when n≥5. This is actually an equation with coefficients in the field \(\mathbb{C}(t_1,t_2,\dots,t_n)\), which is not, of course, a subfield of \(\mathbb C\). As a result Stewart is forced to say a little bit about “the abstract setting” already in section 8.4, way ahead of chapter 16. The main section of this chapter is called “Diet Galois”: in it, Stewart gives a modernized version of the proofs by Ruffini and Abel, which means proving the theorem on “natural irrationalities.” We get several pages of challenging material that is interesting but not really necessary for the ultimate goal.

A final result of the rearrangement is that once we are in the general setting there are many points where the reader is told that the previous arguments are in fact general. The author admonishes us to “seriously work through the material” again rather than “just accept that everything works” (p. 181), but I suspect most readers will adopt the “just accept” strategy.

One valuable addition to the usual bill of fare is chapter 21, which develops parts of Gauss’s cyclotomic theory in order to fill in a gap in the notion of “solvability by radicals” found in most books. The usual definition is that an extension \(L/K\) is a radical extension if \(L=K(\alpha)\) where \(\alpha^m\in K\) for some \(m\). This allows roots of unity “for free” (and the impossibility proofs often begin by assuming they are there), but the “radical” in that case looks like \(\sqrt[m]{1}\), which seems to violate the spirit of the game. So we need a proof that roots of unity can be expressed as radicals in a more natural way, like \[ \zeta_3=\frac{-1+\sqrt{-3}}{2},\qquad \text{and} \qquad \zeta_4=\sqrt{-1}.\] This is provided by Gauss’s theory of cyclotomic periods.

The writing is sometimes quite dense with proofs given without a lot of hand-holding. The pace varies, but it is usually not leisurely. There are many places in which Stewart deviates from usual language and notation. In Stewart’s Field diagrams, for example, the bigger fields are to the right, which seems “sideways” to me; I expected the bigger fields on top. Rings are commutative and unital, but ring homomorphisms to not need to map \(1\) to \(1\). The author notes in passing (p. 5) that nonzero field homomorphisms \(K\to L\) are automatically injective, but he does not give a proof and so does not use that result, preferring to state his theorems in terms of monomorphisms. Thus we get, for example, a proof that the Frobenius automorphism is injective (p. 201). Most notations are defined when they are introduced, but the notation \((m,n)\) for the greatest common denominator is used without comment.

There are more mistakes than I would have expected in a fifth edition. I mention a few that I found particularly striking. The historical introduction refers to the Rubaiyat of Omar Khayyam as a “long poem” (p. xxi) rather than a collection of quatrains. Theorem 2.5 (p. 26) is introduced as an “implication” of the Fundamental Theorem of Algebra, but the proof does not use the FTA. On p. 164, “a group \(G\) is an extension of a group \(A\) by a group \(B\) if \(G\) has a normal subgroup \(N\) isomorphic to \(A\) such that \(G/N\) is isomorphic to \(B\)”; group theorists actually call that an extension of \(B\) by \(A\). On p. 182 the text refers to a “condition \(1\neq 0\) in (M3),” but that condition does not in fact occur in (M3). On p. 271, Theorem 21.9 is said to provide the missing proof of Lemma 21.4, but the theorem is about cyclotomic extensions \(\mathbb{Q}(\zeta)/\mathbb{Q}\) while the Lemma is about the relative case \(\mathbb{Q}(\theta\zeta)/\mathbb{Q}(\theta)\).

While the first edition of Stewart’s book had very little competition in 1973, today there are many good books on Galois Theory. The short books by Artin, Postnikov, and Rotman are not really trying to do the same thing as this book. The major competitors are probably the books by Cox, Weintraub, Tignol, and Bewersdorff. From that list, I would probably go with Cox as a first choice, reflecting the fact that at my college the students in a Galois theory course can be expected to have a fairly good understanding of abstract algebra. Instructors have a wide range of choices.

One should also note, however, that many abstract algebra textbooks already contain a chapter on Galois theory. The question raised by Álvaro Lozano Robledo in his review of Swallow’s Exploratory Galois Theory should also be pondered:

Finally, the reviewer would like to end this note with a personal concern. The undergraduate student (or at least an algebraically oriented student) will have to purchase books which cover groups, rings, fields, Galois theory and so on (other topics, even if not covered in class, might be handy for the student in the future). Should the instructor choose a couple of books which cover (some of) these topics or should the instructor pick a book which contains all of the previous topics (such as Abstract Algebra by Dummit and Foote)?

In the case of Stewart’s book, the reply would be that it contains a lot more than what is usually found in the big textbooks.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.