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 .

 

  1. Determine the number of real and complex embeddings of  (hint: calculus).

 

  1. Find the traces of the first nine powers of  (that is, find ) and use this information to find the discriminant of

 

  1. Find the ramified primes of  and determine how they factor.

 

  1. Compute the Minkowski bound for  and use this information to find the class number and the class structure of

 

  1. 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).

 

  1. 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).