TITLE: Is Galois theory necessary (in a basic graduate algebra course)?
QUESTION [50 upvotes]: 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.)

REPLY [3 votes]: 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!