Math 720
Fall 2000
Final Exam
- 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.
- 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).
- 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 