Math 721
Spring 2001
Exam 2
a)
If
is the splitting
field of
over
and
, show that
acts transitively on the roots of
b)
Show
that if is irreducible of
degree
then
is a transitive
subgroup of
a)
Show
that is not Galois over
b)
Let
be the splitting
field of the polynomial
over
Show that
is Galois over
and compute its
Galois group.
c)
Express
all roots of the polynomial in terms of radicals.
a)
Show
that for some prime
b)
Show
that any element in a finite field can be written as the sum of two squares.