MATH 4121 - Final Exam study guide

The final exam will be Wednesday, May 11 at 10:30AM. You will not be allowed
to use any notes or books during the exam.

Most of the exam will cover material since the midterm exam.
However, I am going to include some Sylow theorem questions on
group theory.  In particular, you should be able to prove that
if P is the only Sylow p-subgroup for some prime p, then P is normal.
Also you should be able to prove that if P is a normal Sylow p-subgroup
for some prime p then P is the only Sylow p-subgroup.
You should also be able to use the counting Sylow theorems to answer
questions like p.333 #5 and 6.

You should also be able to answer problems in field theory like:

p.272 #1-8
p.273 #23
p.291 #1-13
p.292 #19, 24, 29
p.299 #2
p.422 #1-8, 16-18
p.423 #29
p.430 #1-3
p.431 #8
p.434 #1-6

You might be asked to prove Theorem 31.3,

also to show that if x is in E, an algebraic extension of F,
and sigma:E -> E is an automorphism of E fixing F, then
sigma(x) and x are conjugate over F.