Math 421/621
Spring 2000
Final Exam
1.
For
this problem, the notation F[
]=
. Show that the following conditions are equivalent:
a)
F(
)=F[
].
b)
is algebraic over F.
c)
There
is a nonnegative integer N such that every element of F[
] can be written in the form 
(
).
2.
Let
K be a finite, Galois extension field of F (say [K:F]=n). Let G denote the
Galois group of K/F. Define the trace of
to be 
. Also define the norm of
to be 
. Verify the following properties of the norm and trace.
a)
.
b)
.
c)
If
, then 
and 
d)
(G) If
the minimal polynomial of 
in F[x] is of degree
n (say 
find a relationship
between 
and 
and between 
and 
3. Let F be a field whose characteristic is not 2. Let [K:F]=2. Show K is Galois over F (is this true if “[K:F]=2” is replaced by “[K:F]=3”?).
4.
Let
F
K be finite fields of orders 
and 
respectively. Show that m divides n and that K is Galois over
F.
5.
Find
all transitive subgroups of 
(there are two). Find
a cubic polynomial over Q with Galois group 
. (G) Show that the polynomial 
has Galois group
isomorphic to Z3.