Math 720

Fall 2000

Final Exam

 

 

  1. Let  be an integral domain. We define the function  where with  We also declare that .

a)      Show that is a metric on the set  and

b)      Show that  is a dense subspace of

c)      Describe the unit circle in  and Show that if is a field, then the unit circle in R[[x]] is also a group.

d)      Show in general that the units of R[x] and the units of R[[x]] are not isomorphic.

 

  1. Let  be a PID, show that  is a UFD (hint: use the characterization of UFD given in class that every prime contains a principal prime, then in  consider two types of primes…those that contain and those that do not).

 

  1. Let  be a UFD and let be an element of the quotient field of Suppose that is a root of a monic polynomial with coefficients in  (“monic” means that the leading term is 1.) So our polynomial is of the form:

 

 

with

Show that