I seem to have been quite successful in scaring some
students in the last two classes on Galois theory.
My intention was pretty innocent: showing that the
simple problem of finding formulas for the roots of
cubic, quartic, quintic ... polynomials led step by
step to an elaborate abstract theory, namely Galois
theory. If you sat through these classes without
any real idea of what was going on, then you can
imagine how Cauchy must have felt when he saw
Galois' theory exposed in a few densely written
pages: he didn't understand a word. You also should
develop some respect for the genius of Galois who
dreamed up this whole area before he got himself
killed at 20.
What I will not do is cover this material in the final.
The problem is that if I had announced this in advance, then
the classes would have been empty -(.
In any case, those of you with some background in algebra
should have seen how the abstract theory evolved in the
18th century, and that it evolved from very concrete problems,
in our case the solvability of polynomials by radicals.
As for those of you without this background: I hope I
haven't put you off too much, and maybe you'll attend a
course on abstract algebra one day to fill in these gaps.