The ring
of -adic numbers is isomorphic to
(see Exercise 58), so it is not a
field. For example, the element
corresponding to
under this isomorphism has no inverse. (To compute digits
of use the Chinese remainder theorem to find a number that
is modulo and 0 modulo .)
If is prime then
is a field (see Exercise 57).
Since it is a little more
complicated to write -adic numbers down. People typically write
-adic numbers in the form

where
for each .

William Stein
2004-05-06