Is Galois theory necessary (in a basic graduate algebra course)?

Question

By definition, a basic graduate algebra course in a U.S. (or similar) university with a Ph.D. program in mathematics lasts part or all of an academic year and is taken by first (sometimes second) year graduate students who are usually attracted primarily to "pure" mathematics and hope for a career combining some mixture of teaching, problem-solving, basic research. This definition already covers a lot of possibilities, especially if broadened to include those institutions offering only a master's degree. Most users of MO have probably had (or avoided) such an algebra course along the way, beyond an undergraduate introduction.

There is a long "abstract" or "modern" algebra tradition going back to E. Noether and B. van der Waerden, but the steady growth of mathematics has added a huge amount of material to textbooks and has also created too much competition for the time of beginning graduate students. In practice many students can and do bypass algebra at this level. My own sporadic teaching of algebra took place in three quite different departments (Oregon, Courant, UMass) with varying research emphasis on algebraic number theory: the most likely place where mathematicians will really need a lot of Galois theory.

Galois theory has an illustrious history and (to quote Lang) "gives very quickly an impression of depth". It exposes students to real mathematics, combining the study of polynomial rings, fields, and groups in unexpected ways. But it also takes quite a bit of time to develop properly, together with supporting material. And people no longer care as much about solving polynomial equations exactly as about using sophisticated computational methods to estimate roots. In real life the eigenvalues of a big matrix are not estimated by factoring the characteristic polynomial.

Especially in a first semester algebra course taken by a wider range of students, I've found it more rewarding to spend time developing the parallel between finite abelian groups and finitely generated torsion modules over $F[x]$ (unified in the theory of finitely generated modules over PIDs). This is challenging material but gets at some of the canonical form theory for operators in the way most mathematicians should understand it for theory and applications. The minimal polynomial comes into its own here.

Even in a second semester course, where tradition at UMass and many other departments has favored Galois theory, there may be a stronger case to make for teaching basic character theory of finite groups. This too is a meeting ground for many subjects and has even broader applicability than Galois theory when developed into full scale representation theory. (For number theorists, there is the neat proof that degrees of irreducible characters divide the group order.)

Working in algebraic Lie theory and representation theory, subjects unseen by most Ph.D. students, I am especially conscious of choices about which subjects students get exposed to formally. Algebraic and differential geometry often have their own standard (but not first year) courses in departments like UMass, but most people with a Ph.D. in mathematics get by without even those subjects in their background. "What should every mathematician know?" seems more elusive than ever.

Is Galois theory necessary (in a basic graduate algebra course)?

POSTSCRIPT: I appreciate the fact that so many people have actually given the whole issue careful thought, since it bothered me all through my own teaching years. With so little time and so much to learn, choices are inevitable. And it's always easiest to follow the existing course tradition and textbooks. My definition above of "basic graduate course" doesn't fit everywhere, to be sure, but U.S. students usually don't learn much mathematics before that level no matter what their potential is. So the issue won't go away in most U.S. universities that offer advanced work. (It will also continue to be true that most people with a Ph.D. here in "mathematical sciences" will never encounter rigorous Galois theory in courses or in real life.)

Answer 1

I am tempted to stump for the centrality of Galois theory in modern mathematics, but I feel that this subject is too close to my own research interests (e.g., I have worked on the Inverse Galois Problem) for me to do so in a truly sober manner. So I will just make a few brief (edit: nope, guess not!) remarks:

1) Certainly when I teach graduate level classes in number theory, arithmetic geometry or algebraic geometry, I do in practice expect my students to have seen Galois theory before. I try to cultivate an attitude of "Of course you're not going to know / remember all possible background material, and I am more than willing to field background questions and point to literature [including my own notes, if possible] which contains this material." In fact, I use a lot of background knowledge of field theory -- some of it that I know full well is not taught in most standard courses, some of it that I only thought about myself rather recently -- and judging from students' questions and solutions to problems, good old finite Galois theory is a relatively known subject, compared to say infinite Galois theory (e.g. the Krull topology) and things like inseparable field extensions, linear disjointness, transcendence bases....So I think it's worth remarking that Galois theory is more central, more applicable, and (fortunately) in practice better known than a lot of topics in pure field theory which are contained in a sufficiently thick standard graduate text.

2) In response to one of Harry Gindi's comments, and to paraphrase Siegbert Tarrasch: before graduate algebra, the gods have placed undergraduate algebra. A lot of people are talking about graduate algebra as a first introduction to things that I think should be first introduced in an undergraduate course. I took a year-long sequence in undergraduate algebra at the University of Chicago that certainly included a unit on Galois theory. This was the "honors" section, but I would guess that the non-honors section included some material on Galois theory as well. Moreover -- and here's where the "but you became a Galois theorist!" objection may hold some water -- there were plenty of things that were a tougher sell and more confusing to me as a 19 year old beginning algebra student than Galois theory: I found all the talk about modules to be somewhat abstruse and (oh, the callowness of youth) even somewhat boring.

3) I think that someone in any branch of pure mathematics for whom the phrase "Galois correspondence" means nothing is really missing out on something important. The Galois correspondence between subextensions and subgroups of a Galois extension is the most classical case and should be seen first, but a topologist / geometer needs to have a feel for the Galois correspondence between subgroups of the fundamental group and covering spaces, the algebraic geometer needs the Galois correspondence between Zariski-closed subsets and radical ideals, the model theorist needs the Galois correspondence between theories and classes of models, and so forth. This is a basic, recurrent piece of mathematical structure. Not doing all the gory detail of Galois theory is a reasonable option -- I agree that many people do not need to know the proofs, which are necessarily somewhat intricate -- but skipping it entirely feels like a big loss.

Answer 2

In the recent years, the education in mathematics (at the university level) has shifted more toward the applied side. At least in the sense that students have to spend more time on the applied stuff. This of course means that they learn less abstract material. I have taught graduate and undergraduate courses both in the US and in Germany.
My impression is that linear algebra is neglected in the States (one semester course focussing on matrix manipulation), to the extent that graduate students have difficulties with the concept of a basis. On this foundation it might be difficult to teach Galois theory.

However, for mathematicians that plan to do research (get a PhD) it is absolutely necessary to see the merits of an abstract approach. Galois theory not only does this, but also tells you what abstract linear algebra and group theory is useful for. Clearly, groups, fields, vector spaces and polynomials are extremely important concepts in mathematics. After an overemphasis on ways to solve trig integrals in the calculus courses, Galois theory also has the potential to expose students to the true beauty of mathematics (and the fact that many great theories are based on the interaction of different subjects).

Answer 3

I think that, if I got hold of someone who knew no algebra, I would try to teach them rings and modules (together with linear algebra) rather than group theory. The basic reason is that in any kind of "application" (both applied math and applications of algebra to other theoretical math) one always seems to end up linearizing. Even in applications of group theory, especially to field theory, this is true; in fact, groups often come with a natural representation. The most important groups are linear groups.

The other reason for valuing ring theory is that it is more general. The basic group theory course is misleading in its emphasis on convenient consequences of finiteness, and it is also burdened with technical complications related to noncommutativity (of course, the study of finite abelian groups is part of the study of modules, as Jim said. There is a good case for spending time on that in the middle of the beginning of a rings course).

But what about Galois theory? It really ties the whole thing together; using a Galois theory course as an introduction to groups would be fascinating. However, the basic ideas are founded in linear algebra and to do any constructions you need to know basic ring theory. The linear algebra dependence is not strong; it is no stronger than the need to define groups before defining rings (which you really don't need to do), but the idea of having elements of the field act on the field itself as a vector space is not a comfortable one for beginners.

Perhaps viewing Galois theory as an application is itself problematic. It's a higher level theory; as Jim says in the question, people do not solve polynomials symbolically so much in applications. Galois theory is the sort of course you should show people who are on the theoretical track and who have the experience to see it for what it is. However, for a first algebra course I think that rings and modules with linear algebra should come first, followed by groups via some form of representation theory, possibly something with connections to analysis (I was never taught any use of algebra in analysis though I know it exists). Analysis is a big field.

For all its beauty, Galois theory is something of a niche product. There is no right choice here (really, dropping Galois theory seems morally wrong) but it is also worth remembering that it doesn't take a miracle to get students excited about math when they are already in a graduate algebra course. Representation theory is also very appealing (and for a similar reason, in fact), and a solid if utilitarian algebra course is very satisfying.

Answer 4

First, my perspective: at my institution, we teach two streams of undergrad algebra, a standard stream, and an honours-type stream. Both cover some Galois theory, certainly with more being done in the honours stream; in that stream, all the basic theory of finite separable extensions is covered, probably inseparable extensions get a brief mention (char. p is present, but is probably always a little murky compared to char. zero), and insolvability of the quintic is proved.

We also teach a year long (three 10-week (approximately) quarters) first year grad algebra course, which is taken by a mixture of strong undergraduates and first year grad students who haven't passed the prelim in algebra. I think that in the course we presume a background similar to that which we give in our honours-stream undergraduate course.

In the three quarters, the first quarter is group theory and Galois theory. The Galois theory component includes a rapid development/revision of the finite degree case, and a fairly careful treatment of the infinite degree case.

The next two quarters are Wedderburn theory and rep's of finite groups; and then commutative algebra.

If I had to drop one of the three quarters (which is to some extent the context of the question, since it involves passing from quarters to semesters), I would probably drop the first.

My reasoning: Galois theory is vital to algebraic number theory, and useful (sometimes) to algebraic geometry. But I get the feeling that it (like much of algebra, in fact) has the status of esoterica in the larger (pure) mathematical world. Crudely put, people don't use it.

Finite group representation theory, on the other hand, introduces in a simple setting a number of vital concepts which are widely used throughout mathematics. (Wedderburn theory outside the context of group rep'n theory is probably an unnecessary indulgence; it is beautiful, but (for example) the Brauer group is probably even more esoteric than Galois theory. But one can certainly streamline it if necessary, and get pretty straight to the good stuff.)

Commutative algebra is also fairly esoteric, but the linkage to algebraic geometry gives a way of presenting it which adds to the appeal, and in any case, the final quarter/semester probably needs less justification than the first.

One thing that I think is important (which we put into the second quarter) is a good treatment of multilinear algebra; lots of people will need to know this.

Answer 5

I'm learning Galois by rational resolvents, yielding n! distinct values as the roots are permutated. Then the Galois group of a polynomial eqn over Q is generated very wonderfully as a set of permutations that have these two properties regarding rational expressions in the roots. This group is a property of the polynomial equation, with a life of its own. Then the way the group has subgroups as numbers are added to Q is quite extraordinary and equally wonderful. Normality of a subgroup leaps off the page as the Galois group over Q has numbers (or functions) added to it. We see Galois' research actually happening before our eyes. I am a very weak math guy, with a weak undergrad degree, but even I can sense the extraordinary discovery that Galois made when he saw this group emerging. Isn't this work the basis for the whole subsequent course of enquiry regarding "abstraction", as mathematicians looked for invariants and characteristics of entities which were not before suspected or dreamed to exist? I cannot imagine how a student would not be curious as to the provenance of the groovy iso and epi and etc- morphisms and fields and vector spaces et al that he or she learns to be more or less fluent in, for more or less varied reasons.

Answer 6

I am not a graduate student (yet), and don't really know about what first-year grad courses are like, but I don't think that Galois theory should be dropped because "it takes too long to develop", at least not the finite theory.

The only reason why (finite) Galois theory seems to require a lot of time to develop is (I believe) that the standard (I'm inferring from textbooks and course notes I have read) development is backwards. All questions of constructing extension fields (which are the only place where you need ring theory) can be tacitly omitted (or given as homework in a ring theory module), while extending field automorphisms, and the notions of splitting and separable extensions, can be motivated and illustrated by determining the Galois correspondence itself.

A brief outline of the lectures in my head is this. Suppose E is a field. The stabilizer of any group of automorphisms of E must be a subfield of E. Given F a subfield of E, what are necessary and sufficient conditions that F be the stabilizer of a subgroup of automorphisms of E? (why do we care? you're a grad student, that's why you care.)

Clearly, these automorphisms will belong to Aut(E/F), and one immediately derives the necessary conditions that any (irreducible) polynomial f with coefficients in F that has a root in E must have deg f distinct roots in E, since otherwise the product of the (x-r) where r ranges over the distinct roots of the polynomial in E will be fixed by Aut(E:F) and of lower degree than f, thus it will have some coefficient not in F and that coefficient will be fixed by Aut(E/F), which would mean that F is not the stabilizer of Aut(E/F).

The two ways in which polynomial can go bad, then, is that they don't split or have repeated roots, thus motivating the notions of split/normal and separable extensions.

From this perspective, the fundamental theorem of Galois theory says that for finite extensions, the conditions of splitting and separability are in fact sufficient. Here you develop the necessary tools of degree of extension fields, simple extensions, extending automorphisms of intermediate fields and (bless his soul) Emil Artin's linearly independent characters, and you're done (with finite extensions).

Answer 7

Personally, what I feel is that whether the course is required or not really doesn't matter. All that matters is how challenging the student environment is. If you ever feel that they have understood the concepts of groups, rings, and fields clearly, then the next option is Galois theory.

As I said before, it boils down to one thing: how inquisitive the students are in learning the subject.

Answer 8

Double majoring in physics and math, I was undecided about entering graduate school, and if I did, I was anticipating going further into physics instead of math, probably. My junior year abstract algebra course and ESPECIALLY GALOIS THEORY is the reason I chose to apply to math Ph. D. programs. I'm glad I did. 'Nuf said!

Answer 9

I don't really know how to have a clarifying discussion on such topics, but perhaps I have a little distance these days. There is such a topic as "applied algebra", though I suppose it is hardly ever called that: you break down subjects such as algebraic topology, algebraic number theory, algebraic geometry and others, by saying "what here is the algebra that is applied"? This sounds more like what is intended in the question than "group theory for group theorists", ring theory for ring theorists", and so on, counting as "pure algebra". So that Galois theory is not very relevant for algebraic topology, but is very relevant to algebraic number theory and many parts of algebraic geometry. For algebraic topology there is basic material giving access to homological algebra.

Anyway, some effort has to be made to map out areas of mathematics that are active in research terms, and to delineate such algebra as constitutes the prerequisites, to get any relevant answers for graduate education. The approach should be global-to-local. (If it were the other way round, perhaps sheaf theory could be applied.)