Math 772
Spring 2002
Homework 5
Due Friday May 3, 2002. Don’t forget to mark your favorite problem.
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
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.