Proposition 3.4.
Assuming that:
K
is complete with respect to
|
∙
|
Then
(i)
Then
O
K
≅
lim
n
[
]
←
O
K
∕
π
n
O
K
(
O
K
is
π
-
adically complete
)
(ii)
Every
x
∈
O
K
can be written uniquely as
x
=
∑
i
=
0
n
a
i
π
i
,
a
i
∈
A
, where
A
⊆
O
K
is a set of coset representatives for
O
K
∕
π
O
K
.