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,
, and the ring of Witt vectors
Yoshifumi Tsuchimoto
No.02:
Let
be a prime (``base'').
We would like to introduce a metric on
such that
:small
is divisible by powers of
Namely:
DEFINITION 02.1
Let
be a prime number.
 We define a
adic norm
on
as follows.
 We define a
adic distance
on
as follows.
DEFINITION 02.3
A metric space
is said to be
complete
if every Cauchy sequence of
converges to an element of
.
THEOREM 02.4
Let
be a metric space. There exists a complete metric space
with an isometry
such that
is dense
in
. Furthermore,
is unique up to a unique isometry.
DEFINITION 02.5
Let
be a metric space. We call
as in the above theorem
the completion of
.
DEFINITION 02.6
Let
be a prime number. We denote the completion of
by
and call it
the ring of
addic integers.
Thus elements of
are
addic integers.
THEOREM 02.7
has a unique structure of a topological ring.
Namely,
 There exists unique continuous maps
(addition)
and
(multiplication)
which are extensions of the usual addition and multiplication of
.

is a commutative associative ring.
DEFINITION 02.8
Let
be a prime number.
For any sequence
such that
,
we consider a sequence
defined by
Then the sequence
is a Cauchy sequence in
.
We denote the limit of the sequence as
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