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
- 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
- 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).