Math 772
Spring 2002
Exam 2
Due
Wednesday April 17,2002.
In this assignment, we will do an
intensive analysis of a particular ring of integers. For this assignment let
be a root of the
polynomial
Let
and let
be the ring of
integers of
I will give you the
fact that
.
- Determine the number of
real and complex embeddings of
(hint:
calculus).
- Find the traces of the
first nine powers of
(that is, find
) and use this information to find the discriminant of ![](./M772.EX2.S2002_files/image020.gif)
- Find the ramified
primes of
and determine
how they factor.
- Compute the Minkowski
bound for
and use this
information to find the class number and the class structure of ![](./M772.EX2.S2002_files/image025.gif)
- Show that there is an
element in
of norm 9 and an element of norm 27, but no element of
norm 3 (this is of interest since in the set of norms, both 9 and 27 are
irreducible and so you have the factorization
so norm
factorizations are not unique in this case).
- EXTRA CREDIT: Show that
if
is Galois over
and its ring of
integers is a UFD, then its set of norms factor uniquely (extra extra
credit for the converse to this).