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.)
Best Answer
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.