## The Field of -adic Numbers

The ring of -adic numbers is isomorphic to (see Exercise 12), 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 11). 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 2012-09-24