Math 772
Spring 2002
Homework 5
Due Friday May 3, 2002. Don’t forget to mark your favorite problem.
- If
is an odd prime, show that the ring 
contains the 
st roots of unity. What happens in the case 
?
- Fix a
prime integer
a)
Show that any nonzero rational number can be written uniquely
in the form 
with 
an integer and 
relatively prime to 
b)
Show that the function 
(with 
denoting the
rationals and 
denoting the reals)
given by 
where 
if 
and 
Show that 
defines a metric on 
- ( I
have a dream). We may consider a
adic integer as a “formal power series”:
a)
Show that this series converges using the valuation map 
from class.
b)
If we represent the 
adic integer above as a “decimal expansion” 
show that the 
adic integer is a rational number if and only if the decimal
expansion is repeating, or give a counterexample.