Let be a positive integer. Then for any
nonzero rational number there exists a
and integers , , with positive, such that
is exactly divisible by a power of
so for some
is the largest power of
satisfy the conclusion of the lemma.
By unique factorization of integers, there is a smallest multiple
of such that is exactly divisible by . Now apply the
above argument with and replaced by and .
We denote the -adic valuation of by
(Note: Here we are using ``valuation'' in a different way than in the
rest of the text. This valuation is not an absolute value, but the
logarithm of one.)
be a positive integer. For any positive
is as in
-adic valuation of 0
are close in the -adic metric if their
difference is divisible by a large power of . E.g., if then
and are close because their difference is , which
is divisible by a large power of .
the -adic distance
The distance on
defined above is a metric. Moreover,
(This is the ``nonarchimedean'' triangle inequality.)
The first two properties of Definition 14.2.1
immediate. For the third, we first prove that if
Assume, without loss, that
Using Lemma 14.2.5
possibly negative. Then
it follows that
We can finally define the -adic numbers.
is a ring,
but it need not be a field as you will show in Exercises 11 and
12. It is a field if and only if is prime.
has a ``bizarre'' topology,
as we will see in Section 14.2.3.
The set of -adic numbers
, is the
with respect to the metric